Office phone: 510-642-4329.

Office: 923 Evans.

Tentative office hours: Wednesday 1:30-3:00, Thursday 9:30 - 11:00. You are also welcome to make an appointment for another time or just drop in if I'm available. I am very happy to talk about this stuff and try to figure it out with you.

Whew! That sounds awfully ambitious! Well, I have to make it worthwhile for people to show up, because of:

- Morse homology
- Gradient flow lines, the differential, continuation maps, chain homotopies.
- The isomorphism of Morse homology with singular homology.
- Morse theory of circle-valued functions.
- Morse-Bott theory.
- (Depending on time and interest) Morse-theoretic constructions of other topological notions such as Reidemeister torsion and spectral sequences for families.

- Pseudoholomorphic curves in symplectic manifolds
- Basic definitions and lemmas; Gromov-Witten invariants, intersection positivity.
- Some of Gromov's seminal results such as symplectic nonsqueezing and the recognition of R^4.

- Floer theory.
- Basic formalism of symplectic Floer theory of symplectomorphisms and Lagrangian intersections, the quantum product, and the Fukaya category.
- Floer homology of the identity and proof of the Arnold conjecture (modulo some analysis).
- Combinatorial computations of Floer homology on surfaces.
- Gauge-theoretic Floer theory and extended TQFT formalism.
- (Depending on time and interest) more flavors of Floer theory, such as contact homology and Khovanov's categorification of the Jones polynomial, and more applications to symplectic topology.

- Technical tools. (To do all this in detail could take several
semesters, but I want to at least give some introduction.)
- Infinite-dimensional transversality and genericity arguments.
- Compactness arguments.
- Index computations: Maslov index and spectral flow.
- Possibly more technical topics such as (in order of increasing difficulty) orientations, gluing, and virtual cycle techniques.

First installment (ps) Covers the first three or four lectures and gives some references.

Second iteration (ps) Contains corrected version of the first installment together with a new chapter on genericity and transversality.

Third installment (ps) This is just a new chapter on Morse-Bott theory (no corrections to the previous chapters).

Fourth installment (ps) This is a new chapter on Morse theory of circle-valued functions and closed 1-forms.

Please let me know if you find any mistakes in the lecture notes.

** Assignment 1**, due Friday 9/5. (From the lecture notes)
Chapter 2, Exercise 1; Chapter 3, Exercises 1-4.

** Assignment 2**, due Tuesday 9/24. Chapter 4, exercises 1,2;
chapter 5, exercises 1,2,3. Extra credit: the other problems from
chapters 3,4, and 5, and anything you didn't hand in for the first
assignment.

** Assignment 3**, due Tuesday 10/8. Do whichever exercises in
Chapters 6 and 7 are interesting to you. If you haven't seen spectral
sequences before, try computing some examples until you get the hang
of it.

** Final project**. If you want to really learn something, then you
are encouraged, and again this is optional, to investigate in detail
some topic of particular interest to you and then write an expository
article of approximately 10 pages explaining it and/or give a short
talk to the class at the end of the semester.

- "Introduction to symplectic field theory" by Eliashberg, Givental, and Hofer.
- The Floer memorial volume (Birkhauser PM 133).
- Gromov's seminal paper, "pseudoholomorphic curves in symplectic manifolds".
- M. Khovanov, "a categorification of the Jones polynomial".
- The two books by McDuff and Salamon, "J-holomorphic curves and quantum cohomogy" and "introduction to symplectic topology". (A second edition of the former book is in progress, with much more material to be added.)
- Milnor's book "lectures on the h-cobordism theorem", if you can find it.
- The series of recent preprints by Ozsvath and Szabo available from Szabo's web page.
- "Lectures on Floer homology" by D. Salamon.
- The book "Morse homology" by Matthias Schwarz.
- Anything by Paul Seidel.
- The paper "Supersymmetry and Morse theory" by Witten.