Floer Theory, Symplectic Geometry and Mirror Symmetry

辛几何中的DMS-0203593 Yong-Geun oh Floer同调是Floer为了证明Arnold猜想而引入的。Chekanov，oh，PolterovichandSeidel的各种结果已经证明Floer同调是研究辛拓扑的有力工具。在这个项目中，oh建议进一步研究Floer理论的结构和应用，以便更深入地理解Hamilton微分同胚群和Lagrangian子流形或更一般的辛拓扑。通过Fukaya的A-无穷大范畴，Lagrangian子流形的Floer理论也为Kontsevich在Calabi-Yau流形上的同调镜像对称提议提供了一个几何框架。Oh最近与Fukaya、Ohta和Ono的工作提供了几个关键步骤，通过发展定义拉格朗日相交Floer同调的障碍理论来严格构造Fukaya范畴。完整的构建将包括研究奇异拉格朗日子流形、拉格朗日手术及其与Floer同调的关系。Oh建议在Calabi-Yau流形上研究镜像对称性关系中Floer理论的这些新方面。哈密顿形式主义不仅对于解决经典力学中的问题，而且对于将经典力学量子化为量子力学都起着重要的作用。泊松括号是经典力学和量子力学中的重要几何结构，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，即曲线空间上的力学时，相应的相空间和对应于泊松括号的几何结构的描述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。一个是研究哈密顿系统，一种特殊类型的微分方程，以及它们的周期轨道。这本质上是动力学的。二是对拉格朗日子流形的几何和拓扑的研究。这是几何性质的。理解拉格朗日子流形的交理论是辛拓扑的核心。Floer同调是由Floer在80年代末由S引入的，是研究这一交理论的一般机制。Floer理论还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加深对辛拓扑的理解，也旨在通过对镜像对称的研究来理解辛几何和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.