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)

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

• Survey articles
  • C. Wendl, A Beginner's Overview of Symplectic Homology (expanded notes based on the introductory talk of the seminar)
  • A. Oancea, A Survey of Floer Homology for Manifolds with Contact Type Boundary or Symplectic Homology
  • P. Seidel, A biased view of symplectic cohomology
• "Quantitative" symplectic homology (definitions and applications related to symplectic capacities, embedding problems etc.)
  • Sketch in Section 6.6 of H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser 1994.
  • A. Floer and H. Hofer, Symplectic homology I: Open sets in Cⁿ
  • K. Cieliebak, A. Floer and H. Hofer, Symplectic homology II: a general construction
  • A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology I
  • K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology II: Stability of the action spectrum
  • P. Biran, L. Polterovich and D. Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology
  • K. Cieliebak, V. Ginzburg and E. Kerman, Symplectic homology and periodic orbits near symplectic submanifolds
  • D. Hermann, Holomorphic curves and Hamiltonian systems in an open set with restricted contact-type boundary
• Basics of "qualitative" symplectic homology (definitions and computations, relation to the algebraic Weinstein conjecture, wrapped Floer cohomology)
  • C. Viterbo, Functors and computations in Floer homology with applications, I
  • C. Viterbo, Functors and computations in Floer cohomology. Part II
  • M. Abouzaid and P. Seidel, An open string analogue of Viterbo functoriality
• Relations to other invariants (contact homology, Rabinowitz Floer homology, string topology)
  • F. Bourgeois and A. Oancea, Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces
  • F. Bourgeois and A. Oancea, An exact sequence for contact- and symplectic homology
  • F. Bourgeois and A. Oancea, The Gysin exact sequence for S¹-equivariant symplectic homology
  • K. Cieliebak, U. Frauenfelder and A. Oancea, Rabinowitz Floer homology and symplectic homology
  • A. Abbondandolo, M. Schwarz, On the Floer homology of cotangent bundles
  • A. Abbondandolo, M. Schwarz, Floer homology of cotangent bundles and the loop product
  • D. Salamon, J. Weber, Title: Floer homology and the heat flow
• Relations to Lefschetz fibrations, computations, applications to exotic symplectic structures
  • P. Seidel, Symplectic homology as Hochschild homology
  • M. McLean, Lefschetz fibrations and symplectic homology
  • P. Seidel and I. Smith, The symplectic topology of Ramanujam's surface
• Computations via surgery
  • K. Cieliebak, Handle attaching in symplectic homology and the Chord Conjecture
  • F. Bourgeois, T. Ekholm and Y. Eliashberg, Effect of Legendrian Surgery
• Other (recent developments)
  • A. Ritter, Topological quantum field theory structure on symplectic cohomology
  • M. Abouzaid and P. Seidel, Altering symplectic manifolds by homologous recombination

Videos of related talks

• Mark McLean, Symplectic Homology
  A general introduction, including applications to exotic symplectic structures on C²ⁿ and exotic contact spheres.
• Kai Cieliebak, Some remarks on symplectic and contact homology
  Touches upon various topics including the relationship between Symplectic Homology and Contact Homology, Rabinowitz Floer Homology, and (in the last 10 minutes) the isomorphism for subcritical handle attaching and its analogue in Contact Homology.
• Tobias Ekholm, Legendrian contact homology and symplectic homology in dimension four
  Summarizes the main results of the recent preprint by Bourgeois-Ekholm-Eliashberg on Legendrian surgery, including some examples of how to compute Legendrian contact homology.

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