Morse theory and the Thom-Smale-Witten complex

Jonny Evans.

Exercise from point set topology: On a compact space, any continuous function necessarily has a maximum and a minimum.

Exercise from calculus: If F is a differentiable function whose derivative vanishes but whose second derivative is positive/negative then it is a local minimum/maximum.

Morse theory is a vast and sophisticated generalisation of these basic observations: differentiable functions on manifolds can have many kinds of critical points and the Hessian matrix of second derivatives can be used to distinguish several cases according to the signs of its eigenvalues (assuming they're all nonzero, the so-called "Morse" case). Sometimes you are forced to have certain kinds of critical point just for topological reasons. For example, on a torus, any Morse function has at least two saddle points as well as a maximum and a minimum.

You can turn this on its head and use the critical points to study the topology of the manifold. You can also use it to find critical points of functions on infinite-dimensional spaces, otherwise known as solutions to Euler-Lagrange equations (for example geodesics are critical points of the energy functional on loop space).

There are some notes from previous years.