Symplectic and Contact Geometry and Topology

[9971965] eliashberg这个项目的中心主题是辛场论。这是一个有待开发的新领域。它是最近作为接触同调理论的推广而出现的。该项目有几个目标。一是定义接触流形及其legendrian子流形的新不变量，并开发计算这些不变量的工具。第二个目标是探索辛场论中出现的新的代数结构。第三个目标是在辛花理论中构造新的枚举不变量，这些不变量与三维seberg - witten理论密切相关，但可能更容易计算。最后，辛场论应该允许发展具有接触边界的辛流形的Gromov-Witten不变量的相对模拟，从而为计算辛流形的量子上同调提供新的工具。辛几何与拓扑学已被证明在数学和理论物理的不同领域发挥着重要作用。辛几何和拓扑学、哈密顿动力学、低维拓扑学、代数几何、量子场论和其他领域之间相互作用的旧的和新发现的领域产生了许多重要的问题。系统地发展辛场论是当前项目的主要目标，它应该为这些问题提供答案，并为该领域的应用开辟新的领域

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.***