A Morse function on a smooth manifold M is a smooth real-valued function with non-degenerate (and hence isolated) critical points. If the manifold is given a Riemannian metric, the critical points are the stationary points of the associated gradient flow. The critical points and the flow determine the topology of M to a surprising degree. You can see this right away b y pouring hot fudge on your favorite surface and watching it flow; what happens for a torus is very different from what happens for a sphere.

More generally, one can consider infinite-dimensional manifolds such as path/loop-spaces on ordinary Riemannian manifolds; then for a suitable energy function, the critical points are geodesics. This was in fact Morse's original motivation.

The plan for the course is as follows: First we will study basic Morse theory as in part I of the classic text by Milnor: Regular interval theorem, Reeb's theorem, existence of Morse functions, the cell complex defined by a Morse function, and many beautiful applications. An introduction to homology groups will be included.

After that we will survey a series of more advanced topics, to be determined partly by you (see below). Possible such topics include: The h-cobordism theorem and the Poincaré conjecture in dimensions >4, adjoint orbits of compact Lie groups, Morse theory and geodesics, Bott's work on loop spaces of Lie groups and symmetric spaces, Morse-Bott functions, discrete Morse theory, more recent work, e.g. of Cohen-Jones-Segal on Morse theory and classifying spaces of categories, and many more.

In this second part of the course, the idea is that many of these topics would be presented in expository lectures by students. However, I stress that the course is designed to be accessible to anyone who has completed the Manifolds sequence. They will be many opportunities for shorter presentations bases on the first part of the course, and we will all insist that the expository lectures be truly expository and targeted at a general audience.

Besides learning some beautiful mathematics, you will have the opportunity to practice your speaking skills in a friendly, supportive atmosphere. (Don't let you general exam be your first math talk!) Expectations for these talks will vary according to experience. In particular, students who have just completed the Manifolds sequence are strongly encouraged, and reassured that they won't have to prove the Poincaré conjecture. More advanced students will be encouraged to give a complete lecture on a Morse-theoretic topic of their choice.

Prerequisites: Math 544/5/6 (Topology and Manifolds). Besides the basic notions of smooth manifold theory (tangent bundles, smooth maps, inverse function theorem, etc.), the main thing you need is the notion of flow associated to a vector field. It helps to know some homology theory, but I won't assume this.

Text: Morse Theory by John Milnor.