Floerhomology

Wed 9:15-10:45, 11:15-13:45 (Tutorial) RUD25 4.007

Fr 9:15-10:45, RUD 25 3.006

Wed 9:15-10:45, 11:15-13:45 (Tutorial) RUD25 4.007

Fr 9:15-10:45, RUD 25 3.006

We will study the Floer complex of a non-autonomous Hamiltonian on a closed symplectic manifold.

A selection of analytic prerequisites will be discussed. Some details of the construction will be provided

(such as gluing and compactness), others might be explained without proof (regularity, Fredholm property,

transversality) depending on time. The application to the proof of Arnold's conjecture will be explained.

We will conclude with an outlook to other Floer theoretic constructions (e.g. Lagrangian Floer homology and

Fukaya category, symplectic homology and Stein manifolds).

Prerequisites: Notions of calculus of functions with several variables (such as (sub)manifolds, differential forms,

vector fields and their flows), some knowledge on PDE or complex analysis, fundamentals on functionals analysis

(L^p spaces, Banachspaces, bounded linear maps, separation theorem, Banach open mapping theorem)

This is not an exclusive list: If you are missing some knowledge on subjects I will be using ask me for

details on references. A good starter on many of these things are the appendices of (3) and Chapter 2 and 3 in (4)

of the listed literature below.

Tutorials are taught by Mihai Munteanu (firstname dot lastname at hu minus berlin dot de).

Literature: (1) Ana Cannas da Silva: Lectures on Symplectic Geometry, Springer Textbooks

(2) Dietmar Salamon: Lectures on Floerhomology

(3) Michéle Audin, Mihai Damian: Morse theory and Floer homology, Springer Universitext

(4) Dusa McDuff, Dietmar Salamon: J-holomorphic Curves and Symplectic Topology, AMS Publications

(5) Chris Wendl: Lectures on Holomorphic Curves in Symplectic and Contact Geometry,

https://www.mathematik.hu-berlin.de/~wendl/pub/jhol_bookv33.pdf

News: Official date of oral exam (30 minutes): September 10, starting 10 a.m., individual dates can be discussed

Homework Problems

Problem Set 1

Problem Set 2

Problem Set 3

Problem Set 4

Problem Set 5

Problem Set 6

Problem Set 7

Problem Set 8 slightly edited version (06/21)

Problem Set 9

Problem Set 10

Tutorials

notes and comments

Topics

last changes: Fr, 28.6.19, 15:15

A selection of analytic prerequisites will be discussed. Some details of the construction will be provided

(such as gluing and compactness), others might be explained without proof (regularity, Fredholm property,

transversality) depending on time. The application to the proof of Arnold's conjecture will be explained.

We will conclude with an outlook to other Floer theoretic constructions (e.g. Lagrangian Floer homology and

Fukaya category, symplectic homology and Stein manifolds).

Prerequisites: Notions of calculus of functions with several variables (such as (sub)manifolds, differential forms,

vector fields and their flows), some knowledge on PDE or complex analysis, fundamentals on functionals analysis

(L^p spaces, Banachspaces, bounded linear maps, separation theorem, Banach open mapping theorem)

This is not an exclusive list: If you are missing some knowledge on subjects I will be using ask me for

details on references. A good starter on many of these things are the appendices of (3) and Chapter 2 and 3 in (4)

of the listed literature below.

Tutorials are taught by Mihai Munteanu (firstname dot lastname at hu minus berlin dot de).

Literature: (1) Ana Cannas da Silva: Lectures on Symplectic Geometry, Springer Textbooks

(2) Dietmar Salamon: Lectures on Floerhomology

(3) Michéle Audin, Mihai Damian: Morse theory and Floer homology, Springer Universitext

(4) Dusa McDuff, Dietmar Salamon: J-holomorphic Curves and Symplectic Topology, AMS Publications

(5) Chris Wendl: Lectures on Holomorphic Curves in Symplectic and Contact Geometry,

https://www.mathematik.hu-berlin.de/~wendl/pub/jhol_bookv33.pdf

News: Official date of oral exam (30 minutes): September 10, starting 10 a.m., individual dates can be discussed

Homework Problems

Problem Set 1

Problem Set 2

Problem Set 3

Problem Set 4

Problem Set 5

Problem Set 6

Problem Set 7

Problem Set 8 slightly edited version (06/21)

Problem Set 9

Problem Set 10

Tutorials

notes and comments

Topics

- A crash course on symplectic manifolds and Hamiltonian systems (10.4.)

- Problems in Hamiltonian dynamics. The Arnold conjectures on fixed points of Hamiltonian diffeomorphisms (24.4.)
- The Hamiltonian action (26.4.): definition, critical points (26.4.)

- Morse homology (3.5./8.5.)
- Hessian at critical points of the Hamiltonian action, Conley-Zehnder index of periodic solutions (10.5./15.5.)

- Floer's equation and the energy functional (22.5.)

- Floer chain complex (d^2 = 0)
- Floer chain maps associated to homotopy of data
- Homotopy of homotopies: chain maps do not depend on homotopy of data - Floer homology is independent of data

- Proof of Arnold's conjecture: Floer homology is isomorphic to Morse homology (7.6./12.6.)

- Functional analytic setup: The Banach space bundle over the Banach manifold of cylinders, Floer's equation as section of the bundle (12.6./14.6.)
- Fredholm theory and index for the linearization of Floer's equation (17.6-28.6.)
- Transversality (3.7.)
- Convergence to broken Floer trajectories and compactness (5.7.)
- Gluing

last changes: Fr, 28.6.19, 15:15