Leipzig/Berlin Symplectic Homology Learning Seminar

Summer Semester 2010

Time and place

We plan to meet normally every two to three weeks on Thursdays at the Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22, Leipzig. There will also be one meeting at the Humboldt-Universität, Adlershof campus, Rudower Chaussee 25, Berlin. For most sessions there will be two talks, followed by a dinner outing, to which all are welcome.

Schedule of talks (approximate)

Thursday April 29, 2010
MPI, Room A01
  • Chris Wendl: Introduction to Symplectic Homology
    (overview and discussion of topics to be covered)
    expanded notes
Thursday May 20, 2010
MPI, Room A01
  • Felix Schmäschke: Viterbo functoriality
  • Klaus Mohnke: Autonomous Hamiltonians and Morse-Bott moduli spaces
Thursday June 3, 2010
HU Berlin, Rudower Chaussee 25
Room 2.009
  • Klaus Mohnke: Autonomous Hamiltonians and Morse-Bott moduli spaces (conclusion)
  • Matthias Schwarz: Product structures
  • Stephan Mescher: Wrapped Floer cohomology
Thursday June 24, 2010
MPI, Room A01
  • Slava Matveev: Computations via Lefschetz fibrations
  • Oliver Fabert: The exact sequence for symplectic and contact homology
Thursday July 8, 2010
MPI, Room G10
  • Slava Matveyev: Computations via Lefschetz fibrations (conclusion)
  • Alex Krestiachine: The isomorphism for subcritical handle attaching
  • Chris Wendl: The effect of Legendrian surgery

Overview of the seminar

The term symplectic homology (or also cohomology, depending on sign preferences) refers to a family of closely related theories that adapt ideas from Floer homology into the setting of symplectic manifolds with contact type boundary or noncompact symplectic manifolds with cylindrical ends. The original formulation, introduced by Floer, Hofer, Cieliebak and Wysocki in the early 1990s, was of a "quantitative" nature, in that it associated numerical invariants (defined via filtrations on Floer homology) to compact domains in symplectic manifolds, inspired in part by the theory of symplectic capacities. This theory has applications to symplectic embedding questions, and was used for instance to give a symplectic classification of polydisks in the standard Euclidean space, and to show that the interior of a symplectic manifold with contact boundary "sees the boundary" in some sense. In recent years, a more "qualitative" theory has become increasingly popular: first introduced by Viterbo, symplectic homology in this form is an invariant of the symplectic "completion" obtained by adding a cylindrical end to an exact symplectic manifold with contact type boundary. Its definition leads naturally to an algebraic variant of the Weinstein conjecture (and also its proof in many cases), and it has more recently been shown to have deep relations to other invariants of related objects, such as the linearized contact homology of the boundary. Applications include dynamical results related to the Weinstein conjecture, obstructions to exact Lagrangian embeddings, and the existence of exotic Stein structures, among others.

The goal of our seminar will be essentially to learn what symplectic homology is and what it's good for. More specifically, we will begin at the beginning (with the quantitative version) and hope by the end to understand some of the most recent developments, notably the preprint by Bourgeois-Ekholm-Eliashberg computing the effect of Legendrian surgery on symplectic homology, and the work of Seidel, Smith, McLean et al expressing symplectic homology in terms of Lefschetz fibrations.

We assume the target audience for this seminar to be familiar with the main ideas of Hamiltonian Floer homology on closed symplectic manifolds. For other basic notions such as contact manifolds, Stein domains, contact surgery and Lefschetz fibrations, we will attempt to introduce and explain them as needed.

Literature list

The following is a (not very exhaustive) list of articles on symplectic homology, with an attempt to group them into vaguely sensible categories. Some of these are not well suited for seminar talks, but nonetheless contain interesting ideas.

Videos of related talks

Here are some links to the downloadable videos of a few one-hour talks of interest from recent MSRI workshops:

For more information contact me, Chris Wendl by sending e-mail to my surname (at) math (dot) hu-berlin (dot) de