Symplectic Field Theory and related topics

辛场论及相关主题

• 批准号: 0204603
• 负责人: Yakov Eliashberg
• 金额: --
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204603&HistoricalAwards=false
• 关键词: Symplectic; Field; Theory; related; topics

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.

DMS-0204603Yakov Eliashberg 该提案致力于辛场论 (SFT)，其框架最近由 A. Givetal、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、数学中的可积系统理论以及理论物理中的弦理论和镜像对称之间的大型项目。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子流形的不变量； - 在低维拓扑中的应用； - 开发新工具来证明哈密顿系统周期轨道的枚举结果； - 了解格罗莫夫-维滕理论中可积层次结构的出现。 辛场论已经被证明具有重要的应用。特别是，SFT 的思想有助于解决辛几何和哈密顿动力学中几个古老的悬而未决的问题。目前 SFT 的一些研究有望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，而且可能会在理论物理的某些领域（包括弦理论和镜像对称）带来新的应用。除了开发新的应用程序外，该项目的工作还将集中于构建 SFT 的严格基础。