Polyfolds, Fredholm Theory, and Applications

多重折叠、Fredholm 理论及其应用

• 批准号: 0906280
• 负责人: Krzysztof Wysocki
• 金额: $ 13.34万
• 依托单位: Pennsylvania State Univ University Park
• 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 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906280&HistoricalAwards=false
• 关键词: Polyfolds; Fredholm; Theory; Applications

AbstractAward: DMS-0906280Principal Investigator: Krzysztof WysockiIn general terms, the goal of the project is to study analyticalaspects of symplectic geometry and Hamiltonian dynamicalsystems. The overall aim of the project is the development of ageneral framework for studying nonlinear elliptic equationsappearing in symplectic geometry. Many of the problems insymplectic geometry, like the Gromov-Witten theory and theSymplectic Field Theory, are based on the study of the modulispaces of the first order nonlinear elliptic equations. Thesemuduli spaces exhibit lack of compactness, however, they havenontrivial compactification. In the project, Wysocki jointlywith Hofer and Zehnder continues the development of a new generalFredholm theory, which takes place in new smooth spaces calledpolyfolds. This general Fredholm theory has all the properties ofthe classical Fredholm theory, but is more flexible andapplicable to problems with lack of compactness. In the project,the general Fredholm theory on polyfolds is applied to theGromow-Witten theory and to the Symplectic Field Theory. Inanother part of the proposal, we use the theory ofpseudholomorphic curves and the theory of generating functions tostudy multiplicity of closed characteristics on weaklydynamically convex energy surfaces.Symplectic geometry has its origin in classical mechanics. Forexample, the motion of the planetary system can be described by asystem of nonlinear differential equations called Hamiltoniansystems. The flow lines of Hamiltonian systems follow verycomplex patterns. Hence it is of importance to get betterunderstanding of Hamiltonian flows on its energy surfaces. Thedevelopment of the general Fredholm theory on polyfolds will putthe Symplectic Field Theory on solid analytical foundations. Themethods developed in this project should have applications tolarger classes of nonlinear partial differential equations ofrelevance in differential geometry and physics.

AbstractAward：DMS-0906280首席研究员：Krzysztof Wysocki一般来说，该项目的目标是研究辛几何和哈密顿动力系统的分析方面。该项目的总体目标是发展一个研究辛几何中出现的非线性椭圆方程的通用框架。 辛几何中的许多问题，如Gromov-Witten理论和辛场论，都是建立在对一阶非线性椭圆型方程模空间的研究基础上的。这类空间缺乏紧性，但也没有平凡的紧性。 在这个项目中，Wysocki与霍费尔和Zehnder共同继续发展一种新的广义Fredholm理论，这种理论发生在称为polyfolds的新的光滑空间中。这种广义Fredholm理论具有经典Fredholm理论的所有性质，但更灵活，适用于缺乏紧性的问题。在该项目中，关于多重折叠的一般Fredholm理论被应用于Gromow-Witten理论和辛场论。在另一部分中，我们利用伪全纯曲线理论和生成函数理论研究了弱动力凸能量曲面上闭特征的多重性。 例如，行星系统的运动可以用一个叫做哈密尔顿系统的非线性微分方程系统来描述。哈密顿系统的流线遵循非常复杂的模式。因此，深入理解其能面上的哈密顿流是十分重要的.广义Fredholm理论的发展将使辛场论建立在坚实的分析基础上。本计画所发展之方法，应可应用于微分几何与物理中相关之更大类之非线性偏微分方程。