Symplectic Field Theory and related topics

该提案致力于 辛场论（SFT）的框架最近由A。Givental，H.霍费尔和 这个提案的作者。SFT是一个大型项目， 辛几何，哈密顿动力学， 枚举代数几何，低维拓扑，理论 数学中的可积系统，以及弦理论和 理论物理中的镜像对称性。该理论的目标包括：-定义新的，基于SFT的辛和接触不变量 流形及其拉格朗日和勒让德子流形; -应用低维拓扑; -发展新的工具，证明枚举结果 周期轨道 哈密尔顿系统; -理解的外观可积的阶层， Gromov-Witten理论 辛场论已经被证明具有重要的 应用.特别是，思想的SFT有助于解决几个老的悬而未决的问题，辛几何和哈密顿动力学。有希望 目前SFT的一些研究可能会带来一些突破， 在低维拓扑中。理论的进一步发展 也应该带来新的应用不仅在数学，但 可能在理论物理的某些领域，包括弦 理论与镜像对称除了开发新的应用程序外， 该项目下的工作还将集中在建设 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.