- 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.