Symplectic and Contact Geometry and Topology

小行星9971965 这个项目的中心主题是辛场论。这是一个尚待开发的新领域。 它是最近出现的接触同调理论阿萨推广。 该项目有几个目标。 一是定义了切触流形及其Legendrian子流形的新的不变量，并开发了计算这些不变量的工具。 第二个目标是探索新的代数结构出现从辛场论。 第三个目标是在辛Floer理论中构造与三维Seiberg-Witten理论密切相关但计算起来可能更困难的新的计数不变量。 最后，辛场论应该允许发展一个相对类似的Gromov-Witten不变量的辛流形与接触边界，从而提供新的工具，计算辛流形的量子上同调。 辛几何与拓扑在数学和理论物理的各个领域都发挥着重要的作用。 辛几何与拓扑、哈密顿动力学、低维拓扑、代数几何、量子场论和其他领域之间相互作用的新旧领域产生了许多重要问题。 辛场论的系统发展，是当前项目的主要目标，应该为其中一些问题提供答案，并开辟该领域应用的新领域。

9971965Eliashberg The central theme of this project is Symplectic Field Theory.This is a new area that is yet to be developed. It emerged recently asa generalization of Contact Homology Theory. The project has severalgoals. One is to define new invariants of contact manifolds and theirLegendrian submanifolds and develop tools for computing theseinvariants. The second goal is to explore new algebraic structuresemerging from Symplectic Field Theory. The third goal is to constructnew enumerative invariants in Symplectic Floer theory that are closelyrelated to three-dimensional Seiberg-Witten theory but may be easierto compute. Finally, Symplectic Field Theory should permit thedevelopment of a relative analog of Gromov-Witten invariants forsymplectic manifolds with contact boundaries, and thus provide newtools for computing Quantum Cohomology of symplectic manifolds. Symplectic Geometry and Topology has proven to play an importantrole in different areas of Mathematics and Theoretical Physics. The oldand newly discovered areas of interactions between Symplectic Geometryand Topology, Hamiltonian Dynamics, Low-dimensional Topology,Algebraic Geometry, Quantum Field Theory, and other fields generate ahost of important problems. A systematic development of SymplecticField Theory, which is the main goal of the current project, shouldprovide answers to some of these problems, as well as open new areasof applications of the field.***