Math 277: Floer theory

UC Berkeley, Spring 2008


Michael Hutchings. [My last name with the last letter removed] Tentative office hours: Tuesday 1:30-3:30, 923 Evans.

Course outline

The course can roughly be divided into three parts. The first part will cover Morse homology, which is the prototype for all Floer theories. Morse homology recovers the homology of a smooth manifold from dynamical information, namely the critical points of a smooth (Morse) function and the gradient flow lines between them. One can also relate more sophisticated algebraic topological invariants to Morse theory. The second part of the course will introduce pseudoholomorphic curves and Floer homology of symplectomorphisms. The latter is an infinite dimensional generalization of Morse homology which leads to a proof of the Arnold conjecture giving lower bounds on the number of fixed points of generic Hamiltonian symplectomorphisms (and many other applications). The third part of the course will discuss versions of contact homology which give powerful invariants of contact three-manifolds. Some of these are related to the Seiberg-Witten and Ozsvath-Szabo Floer homologies of smooth 3-manifolds.

Unfortunately this subject is highly technical. The course will follow two parallel tracks: one describing the basic formalism, and one explaining how to deal with the technicalities rigorously. The latter track will lag substantially behind the former. I plan to follow up the course with a student seminar in which interested participants can fill in the gaps from the course and explore further.

This course should also provide good preparation for the hot topics week at MSRI from June 9-13 on certain kinds of Floer homology of 3-manifolds and their relations and applications.

There is no textbook for this course, but I will post some salient references below as we go along. I wrote some notes on Morse homology the last time I taught a topics course; these may be useful as an introduction, but beware that they contain some mistakes (which I hope to correct this time around).

Lecture summaries and references