Floer theory seminar
UC Berkeley, Fall 2005
Organized by Michael Hutchings
Course control number 55272
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.
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
Information about the previous incarnation of this seminar may be
- 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:
- 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.