Floer homology in mirror symmetry and in symplectic topology

镜像对称和辛拓扑中的弗洛尔同调

• 批准号: 0904197
• 负责人: Yong-Geun Oh
• 金额: $ 30.03万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904197&HistoricalAwards=false
• 关键词: Floer; homology; mirror; symmetry; symplectic

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.

Abstractaward：DMS-0904197原理研究者：Yong-Geun OhiN项目旨在提供对福卡亚福音歧管的福卡亚类别的描述，以及Calabi-yau的三倍，并建立了基于Fano toric a-Model和TheLandau-Ginzburg B-Modelly flooder flooder flooder floore floore floore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore lore。福卡亚（Ohta Andono）。它还建议通过调查卡拉比同构和entov-Polterovich的准 - 莫尔晶型术语的扩展问题来解决这两个球体（和盘）的更简单问题。此外，PI提议将浮子理论的含义应用于摩托克曼群岛的象征性拓扑，并在汉密尔顿二象形组和Entov-Polterovich的Symphecticquasi-States上构建新的准杂色。 PI预计，规定的研究不仅将提供解决方案的镜像圆环A模型Andlandau-Ginzburg B模型的知识镜对称性，而且还可以使人们对封闭的浮动理论及其在动力学系统和符号拓扑中的应用更深入地了解，以及其在典型的拓扑中，不仅在类似的机构中涉及基本机制，因此可以构成基本的机制，以构成基础，并涉及基本的机制。量子力学。它还在得出从量子场理论到现代弦理论的高素质物理学中的许多基本物理方程式中扮演着动漫的角色。可以对哈密顿形式主义进行的自然空间是象征性的歧管（或更一般的杂种歧管）。在符号歧管中最杰出的研究的几何观察者之一是Lagrangiansubmanifold。例如，“空间中具有零元的粒子的状态”形成了形成symplecticManifold的“空间中粒子的所有可能状态”中的Lagrangian submanifold。了解Llagrangian submanifolds的几何形状与哈密顿流动的动态之间的相互作用是符号拓扑的主题。 PI提出的研究将哈密顿动力学扩展到连续的动力学水平，并解决了从哈米尔顿动力学和镜像对称性引起的各种问题。 它还旨在将研究生和研究人员从其他相关领域的访问权限放到浮动理论的研究中，并镜像对称词，撰写了一本有关浮子同源性及其应用程序的研究生级教科书。