Pseudoholomorphic Curves and Dynamics

AbstractAward：DMS-0606588首席研究员：Krzysztof Wysocki该项目有两个主题：（1）研究辛几何中产生的非线性椭圆方程的一般方法的发展和（2）辛几何的整体方法在动力系统中的应用。该项目的第1部分，与霍费尔和Zehnder联合，为辛场论开发了一个分析框架。它致力于一个一般的非线性Fredholm理论，这发生在新的空间localylvarying尺寸称为polyfolds。该项目的第二个主题是Wysocki及其合作者对有限能量叶理的研究成果。在其中一个子项目中，他将利用有限能量叶理理论证明四维空间中的星形能量表面至少有两个哈密顿周期轨道。在与霍费尔的一个联合子项目中，我们将研究一个长期的区域保持盘映射行为。为了理解这种行为，弗洛尔理论将与有限能叶理论相结合。在该项目的另一部分，Wysocki将使用有限能量叶理来研究四维空间中凸域的辛容量的唯一性。辛几何研究的问题是由天体力学激发的。例如，行星系统的运动可以用一个称为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。该项目为研究这种行为的复杂性提供了新的工具，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Fredholm理论旨在为辛场论提供严格的分析基础。这些思想也适用于数学物理中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.