Floer homology in mirror symmetry and in symplectic topology

奖项：DMS-0904197首席研究员：Yong-Geun Oh这个项目旨在描述Fukaya环面流形和Calabi-Yau三重流形，并基于PI与Fukaya，Ohta Andono共同建立的一般拉格朗日Floer理论，建立Fano Toric A模型和Landau-Ginzburg B模型之间的镜像对应。通过研究Calabi同态和Entov-Polterovich拟同态对哈密顿非同胚群的扩张问题，解决了两个球面(圆盘)的保同胚群的简单性问题。此外，PI还建议将Floer理论的作用机制应用到环面流形的辛拓扑中，并构造了环面流形上的哈密顿微分同态群和Entov-Polterovich辛拟态上的新的拟态射。PI预计，所提出的研究不仅将解决Fano Toric A模型和Landau-Ginzburg B模型的同调镜像对称性，而且将加深对开闭Floer理论及其在动力学系统和辛拓扑中的应用的理解。哈密顿形式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到基础性的作用。它在高能物理中从量子场论到现代弦理论的许多基本物理方程的推导中也起着重要的作用。哈密顿形式主义可以运用的自然空间是辛流形(或者更广泛地说，泊松流形)。辛流形中最著名的几何研究对象之一是拉格朗日子流形；例如，“空间中具有零矩的粒子的状态”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。了解拉格朗日子流形的几何和哈密顿流的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层次，并解决由哈密顿动力学和镜面对称性引起的各种问题。它还旨在通过编写一本关于Floer同调及其在辛几何中的应用的研究生水平的教科书，使研究生和其他相关领域的研究人员更容易进入Floer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.