Polyfolds, Fredholm Theory, and Applications

项目负责人：Krzysztof wysockin总的来说，该项目的目标是研究辛几何和哈密顿动力系统的解析方面。该项目的总体目标是开发研究辛几何中出现的非线性椭圆方程的一般框架。许多非辛几何问题，如格罗莫夫-威滕理论和辛场论，都是基于对一阶非线性椭圆方程的模空间的研究。这些模拟空间缺乏紧性，但具有非平凡紧性。在这个项目中，Wysocki与Hofer和Zehnder一起继续发展一种新的通用fredholm理论，该理论发生在新的光滑空间中，称为多折叠。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。在本课题中，将关于多褶的一般Fredholm理论应用于gromow - witten理论和辛场论。在论文的另一部分，我们利用伪全纯曲线理论和生成函数理论研究了弱动态凸能量曲面上闭特征的多重性。辛几何起源于经典力学。例如，行星系统的运动可以用称为哈密顿系统的非线性微分方程组来描述。哈密顿系统的流线遵循非常复杂的模式。因此，更好地理解其能面上的哈密顿流是很重要的。广义Fredholm多褶理论的发展将为辛场论奠定坚实的分析基础。在这个项目中开发的方法应该应用于微分几何和物理中相关的更大类非线性偏微分方程。

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.