Symplectic Field Theory and related topics

Stanford University, Stanford CA

Investigators

Abstract

DMS-0204603 Yakov Eliashberg The proposal is devoted to Symplectic Field Theory (SFT) whose framework was recently described by A. Givental, H. Hofer and the author of this proposal. SFT is a large project which lies on the borderline between Symplectic Geometry, Hamiltonian Dynamics, Enumerative Algebraic Geometry, Low-dimensional Topology, Theory of Integrable Systems in Mathematics, as well as String Theory and Mirror Symmetry in Theoretical Physics. Among the goals of the theory: - definition of new, SFT-based invariants of symplectic and contact manifolds and their Lagrangian and Legendrian submanifolds; - applications to the Low-dimensional topology; - development of new tools for proving enumerative results about periodic orbits of Hamiltonian systems; - understanding of the appearance of integrable hierarchies in Gromov-Witten theory. Symplectic Field Theory has already proved to have important applications. In particular, the ideas of the SFT were instrumental in the solution of several old outstanding questions in Symplectic Geometry and Hamiltonian Dynamics. There is a hope that some current research in the SFT may bring some breakthrough in the low-dimensional topology. Further development of the theory should also bring new applications not only inside Mathematics, but possibly in some areas of Theoretical Physics, including String Theory and Mirror Symmetry. Besides developing new applications the work under this project will also concentrate on building the rigorous foundations of the SFT.