Pseudoholomorphic Curves and Dynamics

项目负责人：Krzysztof wysocki1 .该项目有两个主题：(1)发展研究非辛几何非线性椭圆方程的一般方法；(2)辛几何全局方法在动力系统中的应用。项目的第一部分，与Hofer和Zehnder合作，为辛场论开发了一个分析框架。它致力于一般非线性fredholm理论，它发生在局部变化维的新空间称为多折叠。该项目的第二个主题是Wysocki及其合作者对有限能量叶的研究成果。在其中一个子项目中，他将使用有限能量叶化理论来证明四维空间中的星形能量面至少携带两个哈密顿周期轨道。区域保留磁盘地图的长期行为将在与Hofer的联合子项目中进行研究。为了理解这种行为，Floertheory将与有限能分形理论相结合。在项目的另一部分，Wysocki将使用有限能量叶研究四维空间凸域辛容量的唯一性。辛几何研究的问题是由天体力学引起的。例如，行星系统的运动可以用一个叫做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。该项目将为研究这种行为的复杂性提供新的工具，并将导致更好地理解星形能量表面上哈密顿流的结构方面。新的广义弗雷德霍姆理论旨在为辛场论提供严密的分析基础。这些思想也适用于数学物理中出现的非线性偏微分方程。

AbstractAward: DMS-0606588Principal Investigator: Krzysztof WysockiThe project has two themes: (1) the development of a generalapproach for studying non-linear elliptic equations arising insymplectic geometry and (2) applications of global methods ofsymplectic geometry to dynamical systems. Part 1 of the project,joint with Hofer and Zehnder, develops an analytical frameworkfor Symplectic Field Theory. It is devoted to a general nonlinearFredholm theory, which takes place on new spaces of locallyvarying dimensions called polyfolds. The second theme of theproject is the outgrowth of research of Wysocki and collaboratorson finite energy foliations. In one of the subprojects he willuse the theory of finite energy foliations to prove thatstar-shaped energy surfaces in four-dimensional space carry atleast two Hamiltonian periodic orbits. A long-term behavior ofarea preserving disk maps will be investigated in a jointsubproject with Hofer. To understand this behavior, the Floertheory will be combined with the theory of finite energyfoliations. In another part of the project, Wysocki will usefinite energy foliations to study the uniqueness of symplecticcapacities of convex domains in four-dimensional spaces. The problems studied in symplectic geometry were motivated bycelestial mechanics. For example, the motion of the planetarysystem can be described by a system of nonlinear differentialequations called Hamiltonian systems. The flow lines ofHamiltonian systems follow very complex patterns. This projectwill provide new tools for studying complexities of this behaviorand will lead to a better understanding of the structural aspectsof Hamiltonian flows on star-shaped energy surfaces. The newgeneral Fredholm theory aims at providing the rigorous analyticalfoundations of Symplectic Field Theory. These ideas should alsobe applicable to nonlinear partial differential equations arisingin mathematical physics.