Towards Relative Symplectic Field Theory

September 24 to September 28, 2007

at the

American Institute of Mathematics, San Jose, California

organized by

Kai Cieliebak, Tobias Ekholm, Yakov Eliashberg, Kenji Fukaya, Dennis Sullivan, and Michael Sullivan

Original Announcement

This workshop CUNY Graduate Center and Stanford Mathematical Research Center (MRC) is to understand the structure of relative Symplectic Field Theory (SFT), discuss and reconcile different versions of its algebraic formalism, and to work towards building rigorous foundations of the theory. There will also be explored applications of relative SFT to symplectic and contact topology, as well as low-dimensional topology.

The goal of the SFT project is to uncover algebraic structures which reflect the topology of the compactified moduli spaces of punctured holomorphic curves in symplectic manifolds with cylindrical ends. Its relative counterpart should describe the topology of the compactified moduli spaces of punctured holomorphic curves with Lagrangian boundary conditions.

Though the SFT project is not yet completed in either absolute or relative case, the involved algebraic structures are much better understood in the absolute case, and the building of the analytic foundations of the theory in the absolute case is well under way. The absolute SFT already led to new invariants of contact and symplectic manifolds and brought many applications to symplectic and contact topology. There were also uncovered deep relations to other subjects such as enumerative algebraic geometry and integrable systems.

On the other hand, relative SFT is still in the period of its infancy. While special cases of the relative theory have been known for a long time, e.g. Floer homology for Lagrangian intersections and Legendrian contact homology, the general formulation of a Relative Symplectic Field Theory has not been yet understood.

On the other hand, in the past few years several new fruitful ideas, and in particular, a link to String Topology were introduced to the subject. As a result, a general picture of Relative SFT is now emerging. By bringing together the researchers involved in these new developments, we intend to reconcile different approaches and establish Relative Symplectic Field Theory in full generality, investigate its relations with Open String Topology, and discuss applications. There will be also discussed the current status of the analytic foundations of the theory, and the remaining necessary steps to complete the project.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A survey article on string topology by Dennis Sullivan.

Papers arising from the workshop:

The role of string topology in symplectic field theory
by  Kai Cieliebak and Janko Latschev