# Math 257C

**Hamiltonian Floer theory **

(Tu-Th 10:30-11:50 @ Herrin T195)

**Lecture Notes**

**Schedule**

Week 1: Stable manifold theorem as an introduction to functional methods

Week 2: Overview of index theory (one class)

Week 3: Spectral flow and indices of operators on cylinders, Maslov index and spectral flow

Week 4: Maslov index and spectral flow (ctd.), Quick completion of the analytic programme for Morse theory

Week 5: Introduction to Hamiltonian Floer theory, energy, asymptotic convergence (statement)

Week 6: Asymptotic convergence (proof), Theorem 1.24 in Salamon

Week 7: Mixed index, transversality, compactness

Week 8: Gluing, orientations

Week 9: Full structure of Hamiltonian Floer theory on general manifolds, higher coherences on continuation maps, Novikov field, homology level operations, symplectic cohomology (no proofs here)

No class during the week of 3-7 June (last week) due to Kylerec

**Sources **

Schwarz - Morse homology (book) W1 (we will go back to Morse theory quite a bit, so this will be used more later)

Atiyah, Patodi, Singer - Spectral asymmetry and Riemannian geometry I, III W2

Kronheimer, Mrowka - Monopoles and Three Manifolds (Sections 14.1, 14.2, 17.1) W3

Salamon, Zehnder - Morse theory for periodic solutions of Hamiltonian systems and the Maslov index W3

Salamon - Lectures in Floer homology (most important reference for the course) W4-8

Floer - The unregularized gradient flow of the symplectic action W4-5

Floer, Hofer, Salamon - Transversality in elliptic Morse theory for the symplectic action W7

Floer, Hofer - Coherent orientations for periodic orbit problems in symplectic geometry W8

Pardon - An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves W9

Floer, Hofer - Floer homology and Novikov rings W9

Seidel - A biased view of symplectic cohomology W9

Last modified Thu, 23 May, 2019 at 23:28