Floer Theory, Symplectic Geometry and Mirror Symmetry

DMS-0203593 Yong-Geun Oh Floer在辛几何中引入了Floer同调，以证明Arnold猜想。Chekanov，Oh，Polterovichand Seidel的许多结果证明了Floer同调是研究辛拓扑的一个有力工具。在本项目中，Oh提出进一步研究Floer理论的结构和应用，以便更深入地理解Hamilton同形群和Lagrange子流形或更一般的辛拓扑。Lagrange子流形的Floer理论，通过福谷的A-无穷范畴，也为Kontsevich关于Calabi-Yau流形的同调镜像定理提供了一个几何框架。Oh最近与福谷、Ohta和Ono一起工作，通过发展一个定义拉格朗日交Floer同调的障碍理论，为严格构造福谷范畴提供了几个关键步骤。完整的构造将涉及奇异拉格朗日子流形的研究，拉格朗日手术和他们的关系到弗洛尔同调。哦建议调查这些新的方面的弗洛尔理论在关系的镜像对称的卡拉比-Hamilton形式不仅对经典力学问题的求解起着重要的作用，而且对经典力学量子化为量子力学也起着重要的作用。 泊松括号是一种重要的几何结构，它通过量子化过程在经典力学和量子力学中起着关键作用。当人们考虑约束系统中的力学时，即，曲空间上的力学，描述相应的相空间和对应于Poisson括号的几何结构需要辛结构和辛流形的概念。辛几何和拓扑学是研究辛流形的学科。在辛几何中，有两个最重要的研究对象。一类是研究Hamilton系统，一种特殊类型的微分方程，以及它们的周期轨道。二是拉格朗日子流形的几何和拓扑研究。理解拉格朗日子流形的交理论是辛拓扑学的核心。Floer同调是Floer在80年代末提出的一种研究交叉理论的通用工具。Floer理论也为物理学家在弦理论中发现的镜像对称现象提供了一个几何框架。Oh提出的研究一方面旨在更深入地理解辛拓扑，另一方面也旨在通过镜像对称的研究来理解辛几何与复几何之间的相互关系。

DMS-0203593Yong-Geun Oh Floer homology in symplectic geometrywas introduced by Floer in an attemptto prove the Arnold conjecture. Various results by Chekanov, Oh, Polterovichand Seidel have proved that the Floer homology is a general powerful tool to investigate symplectic topology.In this project, Oh proposes to further investigate structure and applications of the Floer theory for deeper understanding of the Hamiltonian diffeomorphism group and Lagrangian submanifolds or more generally symplectic topology.The Floer theory of Lagrangian submanifolds, via Fukaya's A-infinity category, also provides a geometric frameworkfor Kontsevich's homological mirror symmetryproposal on Calabi-Yau manifolds. Oh's recent work with Fukaya, Ohta and Ono provides several key steps towards a rigorous construction of Fukaya's category by developing an obstruction theory for defining the Lagrangian intersection Floer homology.Complete construction will involve study of singular Lagrangian submanifolds, Lagrangian surgery and their relations to the Floer homology.Oh proposes to investigate these new aspects of the Floer theoryin relation the mirror symmetry on the Calabi-Yau manifolds.The Hamiltonian formalism playsimportant roles not only for solving problems in classical mechanicsbut also for quantizing the classical mechanics intoquantum mechanics. The Poissonbracket is the crucial geometric structure that plays a key role inthe classical mechanics and the quantum mechanics through thequantization process. When one considers mechanics in a constrained system, i.e., mechanics on a curved space, description of the corresponding phase spaceand the geometric structure corresponding to the Poisson bracketrequires the notion of the symplectic structureand symplectic manifolds. Symplectic geometry and topology is the study of symplectic manifolds. In symplectic geometry, there are two most important objects ofstudy. One is the study of Hamiltonian systems, a special typeof differential equation, and their periodic orbits.This is dynamical in nature. The other is the study of geometryand topology of Lagrangian submanifolds. This is geometric in nature.Understanding the intersection theory of Lagrangian submanifoldsis the core of symplectic topology. Floer homology introduced by Floer in the end of 80's is a general machinery to studythis intersection theory. The Floer theory also provides a geometric framework for the mirror symmetry phenomenon that was discovered by physicists in string theory.Oh's proposed research aims at, on the one hand, deeper understanding ofsymplectic topology, and also aims at understanding inter-relations between the symplectic and the complex geometry via the study of mirror symmetry.