Polyfolds, Fredholm Theory, and Applications

摘要奖：DMS-0906280主要研究员：Krzysztof Wysocki总而言之，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一个研究辛几何中出现的非线性椭圆型方程的通用框架。辛几何中的许多问题，如Gromov-Witten理论和辛场理论，都是基于对一阶非线性椭圆型方程的模空间的研究。这些穆杜里空间缺乏紧致性，然而，它们具有不平凡的紧致性。在这个项目中，Wysocki与Hofer和Zehnder一起继续发展一种新的广义Fredholm理论，该理论发生在称为多折叠的新的光滑空间中。这种广义Fredholm理论具有经典Fredholm理论的所有性质，但更灵活，更适用于紧性不足的问题。在这个项目中，关于多折叠的一般Fredholm理论被应用到Gromow-Witten理论和辛场理论中。在方案的另一部分中，我们利用伪全纯曲线理论和母函数理论来研究弱动力凸能面上闭合特征的多重性。辛几何起源于经典力学。例如，行星系统的运动可以用一个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，更好地理解哈密顿流在它的能面上是很重要的。关于多褶皱的一般Fredholm理论的发展将使辛场理论建立在坚实的分析基础上。本课题所发展的方法应适用于更多类型的与微分几何和物理相关的非线性偏微分方程组。

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.