Symplectic Field Theory and related topics

该提案致力于辛场论（SFT），其框架最近由A. Givental， H. Hofer和该提案的作者描述。SFT是一个介于辛几何、哈密顿动力学、枚举代数几何、低维拓扑学、数学上的可积系统理论以及理论物理上的弦论和镜像对称之间的大型课题。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lagrange和legendian子流形的不变量；-低维拓扑的应用；-开发新的工具来证明关于哈密顿系统周期轨道的枚举结果；-对Gromov-Witten理论中可积层次出现的理解。辛场论已经被证明有重要的应用。特别是，SFT的思想在解决辛几何和哈密顿动力学中几个古老的突出问题方面发挥了重要作用。希望目前在SFT方面的一些研究可以在低维拓扑方面带来一些突破。该理论的进一步发展不仅会在数学领域带来新的应用，还可能在理论物理的某些领域，包括弦理论和镜像对称。除了开发新的应用程序外，本项目下的工作还将集中于建立SFT的严格基础。

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.