Contact Floer Homology

摘要奖：DMS-0072658主要研究员：Alexandre B.Givental在辛拓扑中，所讨论的性质通常是用Floer同调理论和Gromov-Witten不变量的语言来形式化的。所提出的项目是在接触拓扑学的背景下扩展和缠绕这两种形式，其推广与弦理论呈现出自然的关系，并且展示了经典和量子场论的结构是如何从接触Floer同调的弦性质中出现的。该项目被认为是P.I.和Ya的联合工作。埃利亚什贝格(斯坦福大学)等。Hofer(Courant Institute)，是后两位作者几年前开始的更大的项目“辛场论”的一部分。该项目起源于二十世纪初由A.Hurwitz和H.Poincare发起的辛拓扑和接触拓扑学，研究经典力学系统的深刻几何性质。位置和动量坐标的力学概念导致了不对称结构的几何概念，其中力学的哈密顿公式变得非常自然。接触结构是辛结构的奇维对应。除了在辛流形的子空间的研究中出现之外，接触结构在三维空间中也引起了特别的兴趣，其中几何概念阐明了拓扑学中的一些困难问题

AbstractAward: DMS-0072658Principal Investigator: Alexandre B. GiventalIn Symplectic Topology, the properties in question are usuallyformalized in the language of Floer homology theory and Gromov-Witten invariants. The proposed project is to extend andintertwine these two formalisms in the context of Contact Topology.The generalization exhibits natural relationships with the stringtheory and, moreover, shows how the structures of classical andquantum field theory emerge form the stringy properties ofcontact Floer homology. The project is conceived as a joint workof the P.I. with Ya. Eliashberg (Stanford University) andH. Hofer (Courant Institute) and is a part of the larger project"Symplectic Field Theory" started by the latter two authorsseveral years ago.The project originates from Symplectic and Contact Topologylaunched in the beginning of the twentieth century by A. Hurwitzand H. Poincare and studying profound geometrical properties ofclassical mechanical systems. The mechanical notions of positionand momentum coordinates lead to the geometric notion of asymplectic structure, in which the Hamiltonian formulation ofmechanics becomes extremely natural. Contact structures are theodd-dimensional counterparts to symplectic structures. Inaddition to their appearances in the study of subspaces ofsymplectic manifolds, contact structures have been of particularinterest in dimension three, where geometric ideas areilluminating some difficult questions in topology