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\begin{document} \betterposter{
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%%%% Main space
Is the \textbf{lower limb} of the AMOC strictly a \textbf{boundary current} ?

\vspace{0.5em}
r\includegraphics[width=
\textwidth]{na_both.png}
}{

%%%% Bottom space
%% QR code
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\qrcode{figures/qr-code}{figures/smartphone}{ \textbf{Take a picture} to follow \\the project updates. }

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% Author: Freepik
% Retrieved from: https://www.flaticon.com/free-icon/smartphone_65680
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%\compactqrcode{figures/qr-code}{
%\textbf{Take a picture} to
%\\download the full paper
%}
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}{

%%%%%%%% LEFT COLUMN
\title{Unveiling North Atlantic Deep Water pathways using nonlinear dynamics techniques}
\author{P. Miron$^1$, F. J. Beron-Vera$^1$, M. J. Olascoaga$^1$, L. Helfmann$^2$, P. Koltai$^2$ and S. Lozier$^3$} \institution{$^1$University of Miami, Miami} \institution{$^2$Freie Universit\"at, Berlin} \institution{$^3$Duke University, Durham} \institution{\faTwitter\,philippemiron}

\section{\color{miamifushia} Introduction} The North Atlantic Deep water equatorward spreads from high latitudes was expected to be largely contained along the Deep Western Boundary Current.

\vspace{0.2em} New observations from moorings and floats instead showed multiple interior pathways. We propose to analyze deep water circulation from historical Lagrangian observations datasets (Argo and RAFOS floats).

\section{\color{miamifushia} Markov Chain} The information contains in the float trajectories is combined into a transition matrix $P$ representing a Markov Chain of the underlying dynamics. The domain is subdivided into boxes and the entries of $P$ are equal to the conditional transition probabilities between them.
\begin{equation*}
	P_{ij} = \Pr{x_{t+T}\in B_j \mid x_{t}\in B_i}
\end{equation*}

\includegraphics[width=0.9
\textwidth]{pij.png}

%% This fills the space between the content and the logo
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\vfill
\includegraphics[width=
\textwidth]{official-um-rsmas.jpg} }{

%%%%%%%% RIGHT COLUMN

\section{\color{miamifushia} Transition Path Theory} From the constructed Markov Chain on the space $\mathbb{S}$, we are interested in the connection between $A$ and $B$, two subsets of the space $\mathbb{S}$. Using techniques from transition path theory, we can extract how this transition takes place, where trajectories spend more time and what are the main channels. Below in red are the reactive trajectories, segments that connect directly sets $A$ to $B$.

\includegraphics[width=
\textwidth]{reactive-traj.png}

\section{\color{miamifushia} Future work} From surface drifters, deep floats and model outputs, we will investigate the different limbs of the AMOC to characterize the connections between the North Atlantic, where deep water masses are formed, and their sources in the South Atlantic.

\section{\color{miamifushia} References} N. Djurdjevac, Methods for analyzing complex networks using random walker approaches

\vspace{0.15em} L. Helfmann and P. Koltai, Transition path theory for time-dependent dynamics (preprint)

\vspace{0.15em} M. S. Lozier et al., A sea change in our view of overturning in the subpolar North Atlantic

\vspace{0.15em} P. Metzner, Transition path theory for Markov processes

\vspace{0.15em} E. Weinan and E. Vanden-Eijnden, Towards a theory of transition paths. Journal of statistical physics

%% This fills the space
\vfill

%NSF Proposal 1851097
\begin{center}
	\includegraphics[scale=0.15]{nsf_logo.png}
\end{center}
}
\end{document}


\chapter{A bit of manifolds}
Last chapter, we stated Stokes' theorem for cells.
It turns out there is a much larger class of spaces,
the so-called \emph{smooth manifolds}, for which this makes sense.

Unfortunately, the definition of a smooth manifold is \emph{complete garbage},
and so by the time I am done defining differential forms and orientations,
I will be too lazy to actually define what the integral on it is,
and just wave my hands and state Stokes' theorem.

\section{Topological manifolds}
\prototype{$S^2$: ``the Earth looks flat''.}

Long ago, people thought the Earth was flat,
i.e.\ homeomorphic to a plane, and in particular they thought that
$\pi_2(\text{Earth}) = 0$.
But in fact, as most of us know, the Earth is actually a sphere,
which is not contractible and in particular $\pi_2(\text{Earth}) \cong \ZZ$.
This observation underlies the definition of a manifold:
\begin{moral}
	An $n$-manifold is a space which locally looks like $\RR^n$.
\end{moral}
Actually there are two ways to think about a topological manifold $M$:
\begin{itemize}
	\ii ``Locally'': at every point $p \in M$,
	some open neighborhood of $p$ looks like an open set of $\RR^n$.
	For example, to someone standing on the surface of the Earth,
	the Earth looks much like $\RR^2$.

	\ii ``Globally'': there exists an open cover of $M$
	by open sets $\{U_i\}_i$ (possibly infinite) such that each $U_i$
	is homeomorphic to some open subset of $\RR^n$.
	For example, from outer space, the Earth can be covered
	by two hemispherical pancakes.
\end{itemize}
\begin{ques}
	Check that these are equivalent.
\end{ques}
While the first one is the best motivation for examples,
the second one is easier to use formally.

\begin{definition}
	A \vocab{topological $n$-manifold} $M$ is a Hausdorff space
	with an open cover $\{U_i\}$ of sets
	homeomorphic to subsets of $\RR^n$,
	say by homeomorphisms
	\[ \phi_i : U_i \taking\cong E_i \subseteq \RR^n \]
	where each $E_i$ is an open subset of $\RR^n$.
	Each $\phi_i : U_i \to E_i$ is called a \vocab{chart},
	and together they form a so-called \vocab{atlas}.
\end{definition}
\begin{remark}
	Here ``$E$'' stands for ``Euclidean''.
	I think this notation is not standard; usually
	people just write $\phi_i(U_i)$ instead.
\end{remark}
\begin{remark}
	This definition is nice because it doesn't depend on embeddings:
	a manifold is an \emph{intrinsic} space $M$,
	rather than a subset of $\RR^N$ for some $N$.
	Analogy: an abstract group $G$ is an intrinsic object
	rather than a subgroup of $S_n$.
\end{remark}

\begin{example}[An atlas on $S^1$]
Here is a picture of an atlas for $S^1$, with two open sets.
\begin{center}
	\begin{asy}
		size(8cm);
		draw(unitcircle, black+2);
		label("$S^1$", dir(45), dir(45));
		real R = 0.1;
		draw(arc(origin,1-R,-100,100), red);
		label("$U_2$", (1-R)*dir(0), dir(180), red);
		draw(arc(origin,1+R,80,280), blue);
		label("$U_1$", (1+R)*dir(180), dir(180), blue);
		dotfactor *= 2;
		pair A = opendot( (-3, -2), blue );
		pair B = opendot( (-1, -2), blue );
		label("$E_1$", midpoint(A--B), dir(-90), blue);
		draw(A--B, blue, Margins);
		draw( (-1.25, -0.2)--(-2,-2), blue, EndArrow, Margins );
		label("$\phi_1$", (-1.675, -1.1), dir(180), blue);

		pair C = opendot( (1, -2), red );
		pair D = opendot( (3, -2), red );
		label("$E_2$", midpoint(C--D), dir(-90), red);
		draw(C--D, red, Margins);
		draw( (1.25, -0.2)--(2,-2), red, EndArrow, Margins );
		label("$\phi_2$", (1.672, -1.1), dir(0), red);
	\end{asy}
\end{center}
\end{example}

\begin{ques}
	Where do you think the words ``chart'' and ``atlas'' come from?
\end{ques}

\begin{example}
	[Some examples of topological manifolds]
	\listhack
	\begin{enumerate}[(a)]
		\ii As discussed at length,
		the sphere $S^2$ is a $2$-manifold: every point in the sphere has a
		small open neighborhood that looks like $D^2$.
		One can cover the Earth with just two hemispheres,
		and each hemisphere is homeomorphic to a disk.

		\ii The circle $S^1$ is a $1$-manifold; every point has an
		open neighborhood that looks like an open interval.

		\ii The torus, Klein bottle, $\RP^2$, $\CP^2$ are all $2$-manifolds.

		\ii $\RR^n$ is trivially a manifold, as are its open sets.
	\end{enumerate}
	All these spaces are compact except $\RR^n$.

	A non-example of a manifold is $D^n$, because it has a \emph{boundary};
	points on the boundary do not have open neighborhoods
	that look Euclidean.
\end{example}

\section{Smooth manifolds}
\prototype{All the topological manifolds.}

Let $M$ be a topological $n$-manifold with atlas
$\{U_i \taking{\phi_i} E_i\}$.
\begin{definition}
	For any $i$, $j$ such that $U_i \cap U_j \neq \varnothing$,
	the \vocab{transition map} $\phi_{ij}$ is the composed map
	\[
		\phi_{ij} : E_i \cap \phi_i\im(U_i \cap U_j)
		\taking{\phi_i\inv}
		U_i \cap U_j
		\taking{\phi_j} E_j \cap \phi_j\im(U_i \cap U_j).
	\]
\end{definition}
Sorry for the dense notation, let me explain.
The intersection with the image $\phi_i\im(U_i \cap U_j)$
and the image $\phi_j\im(U_i \cap U_j)$ is a notational annoyance
to make the map well-defined and a homeomorphism.
The transition map is just the natural way to go from $E_i \to E_j$,
restricted to overlaps.
Picture below, where the intersections are just the green portions
of each $E_1$ and $E_2$:

\begin{center}
	\begin{asy}
		size(8cm);
		draw(unitcircle, black);
		draw(arc(origin, 1, 80, 100), heavygreen+2);
		draw(arc(origin, 1, -100, -80), heavygreen+2);
		label("$S^1$", dir(45), dir(45));
		real R = 0.1;
		draw(arc(origin,1-R,-100,100), red);
		label("$U_2$", (1-R)*dir(0), dir(180), red);
		draw(arc(origin,1+R,80,280), blue);
		label("$U_1$", (1+R)*dir(180), dir(180), blue);
		dotfactor *= 2;
		pair A = opendot( (-3, -2), blue );
		pair B = opendot( (-1, -2), blue );
		label("$E_1$", midpoint(A--B), dir(-90), blue);
		draw(A--B, blue, Margins);
		draw( (-1.25, -0.2)--(-2,-2), blue, EndArrow, Margins );
		label("$\phi_1$", (-1.675, -1.1), dir(180), blue);

		pair C = opendot( (1, -2), red );
		pair D = opendot( (3, -2), red );
		label("$E_2$", midpoint(C--D), dir(-90), red);
		draw(C--D, red, Margins);
		draw( (1.25, -0.2)--(2,-2), red, EndArrow, Margins );
		label("$\phi_2$", (1.672, -1.1), dir(0), red);

		draw(A--(0.7*A+0.3*B), heavygreen+2, Margins);
		draw(B--(0.7*B+0.3*A), heavygreen+2, Margins);
		draw(C--(0.7*C+0.3*D), heavygreen+2, Margins);
		draw(D--(0.7*D+0.3*C), heavygreen+2, Margins);
		draw(B--C, heavygreen, EndArrow, Margin(4,4));
		label("$\phi_{12}$", B--C, dir(90), heavygreen);
	\end{asy}
\end{center}

We want to add enough structure so that we can use differential forms.

\begin{definition}
	We say $M$ is a \vocab{smooth manifold}
	if all its transition maps are smooth.
\end{definition}

This definition makes sense, because we know what it means
for a map between two open sets of $\RR^n$ to be differentiable.

With smooth manifolds we can try to port over definitions that
we built for $\RR^n$ onto our manifolds.
So in general, all definitions involving smooth manifolds will reduce to
something on each of the coordinate charts, with a compatibility condition.

AS an example, here is the definition of a ``smooth map'':
\begin{definition}
	\begin{enumerate}[(a)]
		\ii Let $M$ be a smooth manifold.
		A continuous function $f : M \to \RR$ is called \vocab{smooth}
		if the composition
		\[ E_i \taking{\phi_i\inv} U_i \injto M \taking f \RR \]
		is smooth as a function $E_i \to \RR$.
		\ii Let $M$ and $N$ be smooth
		with atlases $\{ U_i^M \taking{\phi_i} E_i^M \}_i$
		and $\{ U_j^N \taking{\phi_j} E_i^N \}_j$,
		A map $f : M \to N$ is \vocab{smooth} if for every $i$ and $j$,
		the composed map
		\[ E_i \taking{\phi_i\inv} U_i \injto M
			\taking f N \surjto U_j \taking{\phi_j} E_j \]
		is smooth, as a function $E_i \to E_j$.
	\end{enumerate}
\end{definition}

\section{Regular value theorem}
\prototype{$x^2+y^2=1$ is a circle!}
Despite all that I've written about general manifolds,
it would be sort of mean if I left you here
because I have not really told you how to actually construct
manifolds in practice, even though we know the circle
$x^2+y^2=1$ is a great example of a one-dimensional
manifold embedded in $\RR^2$.

\begin{theorem}
	[Regular value theorem]
	Let $V$ be an $n$-dimensional real normed vector
	space, let $U \subseteq V$ be open
	and let $f_1, \dots, f_m \colon U \to \RR$
	be smooth functions.
	Let $M$ be the set of points $p \in U$
	such that $f_1(p) = \dots = f_m(p) = 0$.

	Assume $M$ is nonempty and that the map
	\[ V \to \RR^m \quad\text{by}\quad
		v \mapsto \left( (Df_1)_p(v), \dots, (Df_m)_p(v) \right) \]
	has rank $m$, for every point $p \in M$.
	Then $M$ is a manifold of dimension $n-m$.
\end{theorem}
For a proof, see \cite[Theorem 6.3]{ref:manifolds}.

One very common special case is to take $m = 1$ above.
\begin{corollary}
	[Level hypersurfaces]
	Let $V$ be a finite-dimensional real normed vector
	space, let $U \subseteq V$ be open
	and let $f \colon U \to \RR$ be smooth.
	Let $M$ be the set of points $p \in U$
	such that $f(p) = 0$.
	If $M \ne \varnothing$ and
	$(Df)_p$ is not the zero map for any $p \in M$,
	then $M$ is a manifold of dimension $n-1$.
\end{corollary}

\begin{example}
	[The circle $x^2+y^2-c=0$]
	Let $f(x,y) = x^2+y^2 - c$, $f \colon \RR^2 \to \RR$,
	where $c$ is a positive real number.
	Note that
	\[ Df = 2x \cdot dx + 2y \cdot dy \]
	which in particular is nonzero
	as long as $(x,y) \ne (0,0)$, i.e.\ as long as $c \ne 0$.
	Thus:
	\begin{itemize}
		\ii When $c > 0$, the resulting curve ---
		a circle with radius $\sqrt c$ ---
		is a one-dimensional manifold, as we knew.
		\ii When $c = 0$, the result fails.
		Indeed, $M$ is a single point,
		which is actually a zero-dimensional manifold!
	\end{itemize}
\end{example}

We won't give further examples
since I'm only mentioning this in passing
in order to increase your capacity to write real concrete examples.
(But \cite[Chapter 6.2]{ref:manifolds} has some more examples,
beautifully illustrated.)

\section{Differential forms on manifolds}
We already know what a differential form is on an open set $U \subseteq \RR^n$.
So, we naturally try to port over the definition of
differentiable form on each subset, plus a compatibility condition.

Let $M$ be a smooth manifold with atlas $\{ U_i \taking{\phi_i} E_i \}_i$.

\begin{definition}
	A \vocab{differential $k$-form} $\alpha$ on a smooth manifold $M$
	is a collection $\{\alpha_i\}_i$ of differential $k$-forms on each $E_i$,
	such that for any $j$ and $i$ we have that
	\[ \alpha_j = \phi_{ij}^\ast(\alpha_i). \]
\end{definition}
In English: we specify a differential form on each chart,
which is compatible under pullbacks of the transition maps.

\section{Orientations}
\prototype{Left versus right, clockwise vs.\ counterclockwise.}

This still isn't enough to integrate on manifolds.
We need one more definition: that of an orientation.

The main issue is the observation from standard calculus that
\[ \int_a^b f(x) \; dx = - \int_b^a f(x) \; dx. \]
Consider then a space $M$ which is homeomorphic to an interval.
If we have a $1$-form $\alpha$, how do we integrate it over $M$?
Since $M$ is just a topological space (rather than a subset of $\RR$),
there is no default ``left'' or ``right'' that we can pick.
As another example, if $M = S^1$ is a circle, there is
no default ``clockwise'' or ``counterclockwise'' unless we decide
to embed $M$ into $\RR^2$.

To work around this we have to actually have to
make additional assumptions about our manifold.
\begin{definition}
	A smooth $n$-manifold is \vocab{orientable} if
	there exists a differential $n$-form $\omega$ on $M$
	such that for every $p \in M$,
	\[ \omega_p \neq 0. \]
\end{definition}
Recall here that $\omega_p$ is an element of $\Lambda^n(V^\vee)$.
In that case we say $\omega$ is a \vocab{volume form} of $M$.

How do we picture this definition?
If we recall that an differential form is supposed to take
tangent vectors of $M$ and return real numbers.
To this end, we can think of each point $p \in M$ as
having a \vocab{tangent plane} $T_p(M)$ which is $n$-dimensional.
Now since the volume form $\omega$ is $n$-dimensional,
it takes an entire basis of the $T_p(M)$ and gives a real number.
So a manifold is orientable if there exists a consistent choice of
sign for the basis of tangent vectors at every point of the manifold.

For ``embedded manifolds'', this just amounts to being able
to pick a nonzero field of normal vectors to each point $p \in M$.
For example, $S^1$ is orientable in this way.
\begin{center}
	\begin{asy}
		size(5cm);
		draw(unitcircle, blue+1);
		label("$S^1$", dir(100), dir(100), blue);
		void arrow(real theta) {
			pair P = dir(theta);
			dot(P);
			pair delta = 0.5*P;
			draw( P--(P+delta), EndArrow );
		}
		arrow(0);
		arrow(50);
		arrow(140);
		arrow(210);
		arrow(300);
	\end{asy}
\end{center}
Similarly, one can orient a sphere $S^2$ by having
a field of vectors pointing away (or towards) the center.
This is all non-rigorous,
because I haven't defined the tangent plane $T_p(M)$;
since $M$ is in general an intrinsic object one has to be
quite roundabout to define $T_p(M)$ (although I do so in an optional section later).
In any event, the point is that guesses about the orientability
of spaces are likely to be correct.

\begin{example}
	[Orientable surfaces]
	\listhack
	\begin{enumerate}[(a)]
		\ii Spheres $S^n$, planes, and the torus $S^1 \times S^1$ are orientable.
		\ii The M\"obius strip and Klein bottle are \emph{not} orientable:
		they are ``one-sided''.
		\ii $\CP^n$ is orientable for any $n$.
		\ii $\RP^n$ is orientable only for odd $n$.
	\end{enumerate}
\end{example}

\section{Stokes' theorem for manifolds}
Stokes' theorem in the general case is based on the idea
of a \vocab{manifold with boundary} $M$, which I won't define,
other than to say its boundary $\partial M$ is an $n-1$ dimensional manifold,
and that it is oriented if $M$ is oriented.
An example is $M = D^2$, which has boundary $\partial M = S^1$.

Next,
\begin{definition}
	The \vocab{support} of a differential form $\alpha$ on $M$
	is the closure of the set
	\[ \left\{ p \in M \mid \alpha_p \neq 0 \right\}. \]
	If this support is compact as a topological space,
	we say $\alpha$ is \vocab{compactly supported}.
\end{definition}
\begin{remark}
	For example, volume forms are supported on all of $M$.
\end{remark}

Now, one can define integration on oriented manifolds,
but I won't define this because the definition is truly awful.
Then Stokes' theorem says
\begin{theorem}
	[Stokes' theorem for manifolds]
	Let $M$ be a smooth oriented $n$-manifold with boundary
	and let $\alpha$ be a compactly supported $n-1$-form.
	Then
	\[ \int_M d\alpha = \int_{\partial M} \alpha. \]
\end{theorem}
All the omitted details are developed in full in \cite{ref:manifolds}.

\section{(Optional) The tangent and contangent space}
\prototype{Draw a line tangent to a circle, or a plane tangent to a sphere.}

Let $M$ be a smooth manifold and $p \in M$ a point.
I omitted the definition of $T_p(M)$ earlier,
but want to actually define it now.

As I said, geometrically we know what this \emph{should}
look like for our usual examples.
For example, if $M = S^1$ is a circle embedded in $\RR^2$,
then the tangent vector at a point $p$
should just look like a vector running off tangent to the circle.
Similarly, given a sphere $M = S^2$,
the tangent space at a point $p$ along the sphere
would look like plane tangent to $M$ at $p$.

\begin{center}
	\begin{asy}
		size(5cm);
		draw(unitcircle);
		label("$S^1$", dir(140), dir(140));
		pair p = dir(0);
		draw( (1,-1.4)--(1,1.4), mediumblue, Arrows);
		label("$T_p(M)$", (1, 1.4), dir(-45), mediumblue);
		draw(p--(1,0.7), red, EndArrow);
		label("$\vec v \in T_p(M)$", (1,0.7), dir(-15), red);
		dot("$p$", p, p, blue);
	\end{asy}
\end{center}

However, one of the points of all this manifold stuff
is that we really want to see the manifold
as an \emph{intrinsic object}, in its own right,
rather than as embedded in $\RR^n$.\footnote{This
	can be thought of as analogous to the way
	that we think of a group as an abstract object in its own right,
	even though Cayley's Theorem tells us that any group is a subgroup
	of the permutation group.

	Note this wasn't always the case!
	During the 19th century, a group was literally defined
	as a subset of $\text{GL}(n)$ or of $S_n$.
	In fact Sylow developed his theorems without the word ``group''
	Only much later did the abstract definition of a group was given,
	an abstract set $G$ which was independent of any \emph{embedding} into $S_n$,
	and an object in its own right.}
So, we would like our notion of a tangent vector to not refer to an ambient space,
but only to intrinsic properties of the manifold $M$ in question.

\subsection{Tangent space}
To motivate this construction, let us start
with an embedded case for which we know the answer already:
a sphere.

Suppose $f \colon S^2 \to \RR$ is a
function on a sphere, and take a point $p$.
Near the point $p$, $f$ looks like a function
on some open neighborhood of the origin.
Thus we can think of taking a \emph{directional derivative}
along a vector $\vec v$ in the imagined tangent plane
(i.e.\ some partial derivative).
For a fixed $\vec v$ this partial derivative is a linear map
\[ D_{\vec v} \colon C^\infty(M) \to \RR. \]

It turns out this goes the other way:
if you know what $D_{\vec v}$ does to every smooth function,
then you can recover $v$.
This is the trick we use in order to create the tangent space.
Rather than trying to specify a vector $\vec v$ directly
(which we can't do because we don't have an ambient space),
\begin{moral}
	The vectors \emph{are} partial-derivative-like maps.
\end{moral}
More formally, we have the following.
\begin{definition}
	A \vocab{derivation} $D$ at $p$ is a linear map
	$D \colon C^\infty(M) \to \RR$
	(i.e.\ assigning a real number to every smooth $f$)
	satisfying the following Leibniz rule:
	for any $f$, $g$ we have the equality
	\[ D(fg) = f(p) \cdot D(g) + g(p) \cdot D(f) \in \RR. \]
\end{definition}
This is just a ``product rule''.
Then the tangent space is easy to define:
\begin{definition}
	A \vocab{tangent vector} is just a derivation at $p$, and
	the \vocab{tangent space} $T_p(M)$ is simply
	the set of all these tangent vectors.
\end{definition}
In this way we have constructed the tangent space.

\subsection{The cotangent space}
In fact, one can show that the product rule
for $D$ is equivalent to the following three conditions:
\begin{enumerate}
	\ii $D$ is linear, meaning $D(af+bg) = a D(f) + b D(g)$.
	\ii $D(1_M) = 0$, where $1_M$ is the constant function on $M$.
	\ii $D(fg) = 0$ whenever $f(p) = g(p) = 0$.
	Intuitively, this means that if a function $h = fg$
	vanishes to second order at $p$,
	then its derivative along $D$ should be zero.
\end{enumerate}

This suggests a third equivalent definition:
suppose we define
\[ \km_p \defeq \left\{ f \in C^\infty M \mid f(p) = 0 \right\} \]
to be the set of functions which vanish at $p$
(this is called the \emph{maximal ideal} at $p$).
In that case,
\[ \km_p^2 = \left\{ \sum_i f_i \cdot g_i
	\mid f_i(p) = g_i(p) = 0 \right\} \]
is the set of functions vanishing to second order at $p$.
Thus, a tangent vector is really just a linear map
\[ \km_p / \km_p^2 \to \RR. \]
In other words, the tangent space is actually the
dual space of $\km_p / \km_p^2$;
for this reason, the space $\km_p / \km_p^2$ is defined as the
\vocab{cotangent space} (the dual of the tangent space).
This definition is even more abstract than the one with derivations above,
but has some nice properties:
\begin{itemize}
	\ii it is coordinate-free, and
	\ii it's defined only in terms of the smooth functions $M \to \RR$,
	which will be really helpful later on in algebraic geometry
	when we have varieties or schemes and can repeat this definition.
\end{itemize}

\subsection{Sanity check}
With all these equivalent definitions, the last thing I should do is check that
this definition of tangent space actually gives a vector space of dimension $n$.
To do this it suffices to show verify this for open subsets of $\RR^n$,
which will imply the result for general manifolds $M$
(which are locally open subsets of $\RR^n$).
Using some real analysis, one can prove the following result:
\begin{theorem}
	Suppose $M \subset \RR^n$ is open and $0 \in M$.
	Then
	\[
	\begin{aligned}
		\km_0 &= \{ \text{smooth functions } f : f(0) = 0 \} \\
		\km_0^2 &= \{ \text{smooth functions } f : f(0) = 0, (\nabla f)_0 = 0 \}.
	\end{aligned}
	\]
	In other words $\km_0^2$ is the set of functions which vanish at $0$
	and such that all first derivatives of $f$ vanish at zero.
\end{theorem}
Thus, it follows that there is an isomorphism
\[ \km_0 / \km_0^2 \cong \RR^n
	\quad\text{by}\quad
	f \mapsto
	\left[ \frac{\partial f}{\partial x_1}(0),
		\dots, \frac{\partial f}{\partial x_n}(0) \right] \]
and so the cotangent space, hence tangent space,
indeed has dimension $n$.

%\subsection{So what does this have to do with orientations?}
%\todo{beats me}

\section\problemhead
\begin{problem}
	Show that a differential $0$-form on a smooth manifold $M$
	is the same thing as a smooth function $M \to \RR$.
\end{problem}
\todo{some applications of regular value theorem here}