These are the notes from the twenty minute talk I'm going to give at

the UCL Geometry and Topology Open Day for prospective PhD students

(November 14th 2012). For those who are interested in reading more,

check out Milnor's book on Morse theory and Atiyah's early survey

paper on Floer theory.

Given a function \(F\colon\mathbf{R}^n\to\mathbf{R}\), the most

interesting points we can study are its critical points, in other

words the points where all the partial derivatives vanish:

\[\frac{\partial F}{\partial x_1}=\frac{\partial F}{\partial

x_2}=\cdots=\frac{\partial F}{\partial x_n}=0\]

For example, if \(F(x,y)=x^2-y^2\) so that \(\partial F/\partial x=2x\),

\(\partial F/\partial y=-2y\), then the only critical point is

\(x=y=0\). If we draw level sets of \(F\) (the contours of \(F\) considered

as a height function) then the level sets undergo a significant change

when we look a little below and a little above the critical

height 0. This particular critical point is called a saddle point (for

obvious reasons when you look at a picture of its graph).

This is the first indication of a deep relationship between the

critical points of a function and the topology (or shape) of the

domain of the function. Of course we could consider functions on more

interesting spaces than \(\mathbf{R}^n\). For instance, the height

function on a sphere has two critical points (a maximum at the top, a

minimum at the bottom) and the height function on a torus has four (a

max, a min and two saddle points). You can tell what kind of critical

point you have by looking at the number of downward directions of the

function at the critical point: at a maximum on the torus there are

two downward directions, at a saddle there is one downward direction

and at a minimum there are none. To be more precise we need to recall

the second derivative test which says that the Taylor expansion of a

function around a critical point is dominated by its Hessian

matrix. Negative eigenvalues of the Hessian mean downward directions

(eigendirections) of the function, positive eigenvalues mean upward

directions. Let's define the index of a critical point to be the

number of negative eigenvalues of the Hessian at that point.

The deep relationship between critical points and topology is due to

Morse:

Suppose that \(F\colon M\to\mathbf{R}\) is a function on a manifold

\(M\) and that, for all critical points, the Hessian matrix has no

zero eigenvalues (nondegeneracy). Then there are at least \(b_i\)

critical points with index \(i\) where \(b_i\) is a purely topological

quantity called the $i$th Betti number, counting the number of “linearly independent \(i\)-dimensional holes”.

For instance, in a torus there are two “linearly independent

1-dimensional holes” and in our example we had two critical points of

index 1 (saddles). Conversely, with a little more information about

the gradient flowlines of the function, one can say something about

the topology of M from data about the function F and this is a very

fruitful piece of topology which you can read about in Milnor's

classic textbook on Morse theory.

More interestingly, one can study functions on infinite-dimensional

spaces \(M\). For example, \(M\) might be the space of paths

\(\gamma\colon[0,1]\to K\) in a Riemannian manifold \(K\) with fixed

endpoints and the function \(F\) might be the length functional

\[F(\gamma)=\int_0^1|\dot{\gamma}(t)|dt\] The critical points of this

functional are familiar: they are the solutions to the Euler-Lagrange

equation, in other words the geodesic paths. In this case you can

still make sense of the index of a critical point – though \(M\) is

infinite-dimensional, there are only a finite-dimensional space of

downward directions. Indeed when \(\gamma\) is length-minimising there are

no downward directions! Using this kind of Morse theory on the

loopspaces of certain homogeneous spaces, Bott was able to prove his

remarkable periodicity theorem for the homotopy groups of the

orthogonal and unitary groups.

Yet more interesting is the case when the index becomes infinite, the

case when your critical points have infinitely many upward and

infinitely many downward directions. This is called Floer theory or

semi-infinite Morse theory, and is significantly harder. Let me give

you an example.

Let \(H\colon\mathbf{R}^3\to\mathbf{R}\) be a function. On the space of

\(2\pi\)-periodic functions \(\mathbf{R}\to\mathbf{R}^2\), in coordinates

\(t\mapsto (p(t),q(t))\), you can define the symplectic action

functional

\[(p,q)\mapsto\int_0^{2\pi}(p(t)\dot{q}(t)-H(t,p(t),q(t)))dt\] The

Euler-Lagrange equations are \[\frac{\partial L}{\partial

p}=\frac{d}{dt}\frac{\partial L}{\partial\dot{p}},\qquad\frac{\partial

L}{\partial q}=\frac{d}{dt}\frac{\partial L}{\partial\dot{q}}\] which

become \[\dot{q}=\frac{\partial H}{\partial

p},\qquad\dot{p}=-\frac{\partial H}{\partial q}\] These you may

recognise as Hamilton's equations of motion for the Hamiltonian

function \(H\). So a critical point of the symplectic action functional

is a \(2\pi\)-periodic orbit of this Hamiltonian system. Let me convince

you that it has infinite index, at least when \(H\equiv 0\). Let

\(p(t)=\sum_{k=-\infty}^{\infty}p_k^{ikt}\) and

\(q(t)=\sum_{k=-\infty}^{\infty}q_ke^{ikt}\) where \(p_{-k}=\bar{p}_k\)

and \(q_{-k}=\bar{q}_k\) are the Fourier coefficients. Then

\[F(p+\delta,q+\epsilon)=\sum_k\sum_{\ell}(p_k+\delta_k)i\ell(q_\ell+\epsilon_{\ell})\int_0^{2\pi}e^{i(k+\ell)t}dt\]

The integral is nonzero only when \(k=-\ell\), when it is \(\pi\), so this

becomes

\[F(p+\delta,q+\epsilon)=\sum_{k=-\infty}^{\infty}k\pi(p_k+\delta_k)(q_k+\epsilon_k)\]

This is

\[F(p+\delta,q+\epsilon)=\pi\sum_{k=-\infty}^{\infty}(kp_kq_k+k(p_k\epsilon_k+q_k\delta_k)+k\delta_k\epsilon_k)\]

The Hessian is (twice) the quadratic part of this. If we define

\[\mathbf{H}=\pi\left(\begin{array}{ccccccc} \ddots & 0 & 0 & 0 & 0 &

0 & 0\\ 0 & -2 &0 &0 & 0 & 0 & 0 \\ 0 & 0 & -1&0 &0 & 0 & 0 \\ 0 & 0 &

0 & 0 &0 &0 & 0 \\ 0 & 0 & 0 & 0 &1 &0 & 0 \\ 0 & 0 & 0 & 0 &0 &2 & 0

\\ 0 & 0 & 0 & 0 &0 &0 & \ddots \end{array}\right)\] then the Hessian

is \[\left(\begin{array}{cc}\delta

&\epsilon\end{array}\right)\left(\begin{array}{cc}0 &

\mathbf{H}\\ \mathbf{H} &

0\end{array}\right)\left(\begin{array}{c}\delta\\ \epsilon\end{array}\right)\]

and we see that the Hessian is an infinite matrix with infinitely many

positive and negative eigenvalues!

Floer's idea was to prove some kind of invariance of this theory under

deformations of \(H\), which allowed him to compare the number of critical

points (periodic orbits) for an arbitrary \(H\) to the number of periodic

orbits for a particular H he could understand. He used this to prove

the Arnold conjecture: a highly nontrivial lower bound on the number

of periodic orbits.

Morse theory is related by Morse's theorem to the topology of a

manifold. One exciting open problem is to understand the topology of “semi-infinite cycles” and their intersection theory, which is what

Floer's theory seems to capture. A slighty more precise version of

this open problem goes by the name of the Atiyah-Floer conjecture. You

can read about it here:

- M. F. Atiyah, “New invariants of 3- and 4-dimensional manifolds” in

The Mathematical Heritage of Hermann Weyl (Proceedings of Symposia

in Pure Mathematics, Volume 48) 1988 p. 285 – 300