Floer theory seminar

UC Berkeley, Fall 2005

Organized by Michael Hutchings

Course control number 55272

Outline

This seminar will discuss various aspects of Floer theory, with an emphasis on those versions of Floer theory that count pseudoholomorphic curves in symplectic 4-manifolds. The goal will be to learn something about both analytical foundations and computational techniques. I will begin by giving a few talks on some topics that I am interested in, and other participants are also welcome to give talks. There will also be some extra background lectures.

Spacetime coordinates

The main talks will take place on Thursdays from 5:10 to 6:00 PM in room 891. There will also be a parallel series of background lectures on Mondays from 3:10 to 4:00 PM in room 959, at least at the beginning.

Schedule of talks

Thursday 9/8 at 5 PM in room 891. (M. Hutchings) Introduction to three-dimensional Floer theories. There are a number of different ways to define Floer homology for a three-manifold, including instanton Floer homology, Seiberg-Witten Floer homology, Ozsvath-Szabo invariants, symplectic field theory, and embedded contact homology. I will give an introduction to these theories and how they might be related (or not). Subsequent talks will give more details about the last two theories.
Monday 9/12 at 3 PM in room 959. (M. Hutchings) Morse homology. All versions of Floer theory can be regarded as infinite dimensional variants of one prototype. The prototype is Morse homology on a finite dimensional manifold, which will be reviewed in this talk. References mentioned in the talk:
- My lecture notes on Morse homology (beware that these contain some mistakes).
- M. Schwarz, Morse homology, Birkhauser, 1993.
- E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982), 661--692.
Thursday 9/15 at 5 PM in room 891. (M. Hutchings) Review of symplectic field theory. The symplectic field theory of Eliashberg, Givental, and Hofer defines Floer theory type invariants of a contact manifold by counting J-holomorphic curves in the symplectization. In addition to providing new information about contact manifolds, these invariants naturally enter into gluing formulas for Gromov-Witten invariants. This talk will review some of the most basic ideas of this theory. Some references:
- Y. Eliashberg, A. Givental, H. Hofer Introduction to symplectic field theory.
- F. Bourgeois, Introduction to contact homology.
There will be no background lecture on Monday 9/19, but there might be one the following week. If you have suggestions for topics for background lectures, please let me know.
Thursday 9/22 at 5 PM in room 891. (M. Hutchings) Review of symplectic field theory, part 2. We will continue to discuss cylindrical contact homology, which counts J-holomorphic cylinders in the symplectization of a contact manifold. We will then introduce the formalism for counting more general J-holomorphic curves than cylinders.
Thursday 9/29 at 5 PM in room 891. (M. Hutchings) Embedded contact homology, part 1. We explain how to define a variant of symplectic field theory for contact 3-manifolds that counts certain embedded pseudoholomorphic curves in the symplectization, and is conjecturally isomorphic to the Ozsvath-Szabo and Seiberg-Witten Floer homologies. Some papers about this may be found here.
Thursday 10/6 at 5 PM in room 891. (M. Hutchings) Examples of embedded contact homology. We discuss how to compute the embedded contact homology of T^3. (joint work with M. Sullivan)
Thursday 10/13: No Meeting.
Thursday 10/20 at 5 PM in room 891. (M. Hutchings) Morse-Bott theory revisited. We introduce the recent approach to Morse-Bott theory due to Frederic Bourgeois.
Monday 10/24 at 3 PM in room 959. (D. Farris) The Maslov and Conley-Zehnder indices. The Maslov indices are closely related invariants of loops and paths in the symplectic group or in the Lagrangian Grassmannian. I will define the different variants of them and describe their appearance in situations such as Lagrangian immersion and the Conley-Zehnder index of periodic orbits, which appears in index formulae for the dimension of Floer-theoretic moduli spaces of holomorphic curves.
Thursday 10/27 at 5 PM in room 891. (M. Hutchings) Branched covered cylinders and Feynman diagrams. To show that d^2=0 in embedded contact homology, one needs to glue two pseudoholomorphic curves together by inserting a branched cover of a cylinder in between them. The number of such gluings is given by the number of zeroes of a section of an obstruction bundle over the moduli space of branched covers. This can be computed as a certain sum over labeled trees. (joint work with C. Taubes)
Monday 10/31 at 3 PM in room 959. (D. Farris) The Maslov index and moduli spaces of holomorphic cylinders. We will recall the definition of the Maslov index and explain why it computes the dimensions of moduli spaces of holomorphic cylinders in Floer theory.
Thursday 11/4: No Meeting. (Bowen lecture by John Conway)
Further talks TBA.

Information about the previous incarnation of this seminar may be found here.