Floer homology in mirror symmetry and in symplectic topology

摘要：本项目旨在基于PI与Fukaya， Ohta和ono共同建立的一般拉格朗日Floertheory，给出环流形和Calabi-Yau三倍的Fukaya范畴的描述，并建立Fano环a模型与andau- ginzburg b模型之间的镜面对应关系。通过研究Calabi同态和Entov-Polterovich拟同态对哈密顿同态群的扩展问题，提出了两个球（和圆盘）的保面积同态群的简单性问题。此外，PI还提出将花理论的机制应用到环流形的辛拓扑中，并在环流形上构造哈密顿和微分同构群上的新拟态和Entov-Polterovich的辛拟态。PI期望所提出的研究不仅可以解决Fano环a模型和landau - ginzburg b模型的同调镜像对称性，而且可以加深对开闭花理论及其在动力系统和辛拓扑中的应用的理解。哈密顿形式主义不仅对解决经典力学中的问题，而且对将经典力学转化为量子力学起着根本性的作用。从量子场论到现代弦理论，它在高能物理中许多基本物理方程的推导中也起着重要作用。可以运用哈密顿形式主义的自然空间是辛流形（或者更一般地说是泊松流形）。辛流形中最杰出的几何研究对象之一是拉格朗日子流形；例如，“空间中具有零动量的粒子的状态”在“空间中粒子的所有可能状态”中形成拉格朗日子流形，从而形成辛流形。理解弗拉格朗日子流形几何与哈密顿流动力学之间的相互作用是辛拓扑的核心主题。PI提出的研究旨在将哈密顿动力学扩展到连续动力学的水平，并解决哈密顿动力学和镜像对称引起的各种问题。它还旨在通过编写关于花同调及其在辛几何中的应用的研究生水平教科书，方便研究生和其他相关领域的研究人员进入花理论和镜像对称的研究。

AbstractAward: DMS-0904197Principal Investigator: Yong-Geun OhThis project aims at providing a description of Fukaya categoriesof toric manifolds and of Calabi-Yau threefolds, and establishingthe mirror correspondence between the Fano toric A-model and theLandau-Ginzburg B-model based on the general Lagrangian Floertheory established by the PI jointly with Fukaya, Ohta andOno. It also proposes to solve the simpleness problem of the areapreserving homeomorphism group of the two sphere (and the disc)by investigating the extension problem of Calabi homomorphismsand Entov-Polterovich's quasi-morphisms to the Hamiltonianhomeomorphism group. In addition, the PI proposes to apply themachinery of Floer theory to symplectic topology of toricmanifolds and construct new quasi-morphisms on the Hamiltoniandiffeomorphism group and Entov-Polterovich's symplecticquasi-states on toric manifolds. The PI anticipates that theproposed research will not only provide solution to thehomological mirror symmetry of Fano toric A-model andLandau-Ginzburg B-model but also lead to deeper understanding ofopen-closed Floer theory and its applications to dynamicalsystems and symplectic topology.The Hamiltonian formalism played a fundamental role not only forsolving problems in classical mechanics but also for transformingthe classical mechanics into quantum mechanics. It also plays animportant role in deriving many basic physical equations in highenergy physics ranging from quantum field theory to modern stringtheory. The natural space where the Hamiltonian formalism can beexercised is the symplectic manifold (or more generally thePoisson manifold). One of the most distinguished geometricobjects of study in symplectic manifold is the Lagrangiansubmanifold; For example, `the state of a particle with zeromomentum in space' forms a Lagrangian submanifold in `allpossible states of a particle in space' which forms a symplecticmanifold. Understanding the interplay between geometry ofLagrangian submanifolds and dynamics of Hamiltonian flows is thecore theme of symplectic topology. The PI's proposed researchaims at extending the Hamiltonian dynamics to the level ofcontinuous dynamics and solving various problems arising fromHamiltonian dynamics and mirror symmetry. It also aims at easingthe access of graduate students and researchers from otherrelated fields into the study of Floer theory and mirror symmetryby writing a graduate level textbook on Floer homology and itsapplications to symplectic geometry.