Symplectic Field Theory and related topics

DMS-0204603 Yakov Eliashberg本提案致力于辛场论(SFT)，其框架最近由A.Givental、H.Hofer和本提案的作者描述。SFT是一个介于数学中的辛几何、哈密顿动力学、列举代数几何、低维拓扑、可积系统理论以及理论物理中的弦理论和镜面对称之间的大项目。该理论的目标包括：-定义新的基于SFT的辛流形和接触流形及其拉格朗日和勒让德日子流形；-应用于低维拓扑学；-开发新的工具来证明关于哈密顿系统的周期轨道的计数结果；-理解Gromov-Witten理论中的可积族的出现。辛场理论已经被证明具有重要的应用。特别是，SFT的思想对解决辛几何和哈密顿动力学中的几个古老的未决问题起到了重要作用。目前在SFT中的一些研究有望为低维拓扑带来一些突破。该理论的进一步发展不仅应该在数学领域带来新的应用，而且可能在理论物理的某些领域带来新的应用，包括弦理论和镜面对称性。除了开发新的应用程序外，该项目下的工作还将集中于为SFT奠定坚实的基础。

DMS-0204603Yakov 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.