Symplectic Topology, Symplectic Submanifolds and Floer Theory

摘要奖：DMS-0405994主要研究员：Ely Kerman这项建议由三个项目组成，涉及辛流形的各种不变量与其支持的哈密顿流的周期轨道之间的关系。Kerman最近的工作表明，其中一个不变量，Hofer-Zehnder容量，对于某些辛子流形的管状邻域是有限的。利用Biran的分解定理，给出了紧致Kahler流形的几种新的辛交现象。第一个项目的目标是研究这些新的交集结果，这些结果表明紧致Kahler流形的许多基本辛性质是由Biran分解决定的。第二个项目是与V.L.金兹堡和B.Gurel的联合努力。它涉及到哈密顿Floer同调的广义形式的构造，其中不同同伦类中的周期轨道被允许通过计算带有穿孔的扰动全纯曲线的广义Floer微分来相互作用。这一建构是由Eliashberg、Givental和Hofer的辛场理论推动的。生成理论还应该具有丰富的代数结构，以及各种应用，包括弱正合辛流形的Hofer-Zehnder容量的新计算。第三个项目是证明一个猜想的程序，该猜想断言在哈密顿量的非退化辛临界子流形附近的所有水平集上存在周期轨道。这是Arnold的一些类似猜想的推广，这些猜想涉及带电粒子在磁场中运动的周期轨道。第一步是为基本变分原理构造Floer型不变量。一旦它被严格定义，这应该会很快导致许多新的存在结果。人们还希望这个不变量可以用来扩充辛场理论，它允许人们沿着某些非接触型超曲面分裂辛流形。哈密顿流被用来模拟许多重要的能量守恒的物理系统。这样的系统包括行星和卫星在相互引力下运动，带电粒子在电磁场中运动，以及不可压缩的理想流体的流动。这些情绪通常是相当复杂的，开始了解它们的全球行为的一种方法是寻找重复的模式，即周期轨道。虽然大多数哈密顿流都有许多周期轨道，但要确定它们在固定能级上的存在性通常是一个困难的问题。这个问题是哈密顿流研究中的一个中心主题，在现代，已被证明与定义流的空间的形状密切相关。该提案中的项目研究了这种关系的各个方面。在前两个项目中，我们使用哈密顿流来定义和计算辛不变量。最后一个项目涉及构造一个新的辛不变量，它应该导致描述带电粒子在磁场中运动的哈密顿流的周期轨道的新的存在结果。

AbstractAward: DMS-0405994Principal Investigator: Ely KermanThis proposal is comprised of three projects which concern therelation between various invariants of a symplectic manifold andthe periodic orbits of the Hamiltonian flows which it supports.Recent work by Kerman shows that one of these invariants, theHofer-Zehnder capacity, is finite for tubular neighborhoods ofcertain symplectic submanifolds. Using a decomposition theorem ofBiran, this implies several new kinds of symplectic intersectionphenomena for compact Kahler manifolds. The goal of the firstproject is to study these new intersection results which suggestthat many basic symplectic properties of a compact Kahlermanifold are determined by the Biran decompositions itadmits. The second project is a joint effort with V.L. Ginzburgand B. Gurel. It involves the construction of a generalizedversion of Hamiltonian Floer homology in which periodic orbits indifferent homotopy classes are allowed to interact via ageneralized Floer differential that counts perturbed holomorphiccurves with punctures. The construction is motivated by theSymplectic Field Theory of Eliashberg, Givental and Hofer. Theresulting theory should also have a rich algebraic structure, aswell as a variety of applications including new calculations ofthe Hofer-Zehnder capacity for weakly-exact symplecticmanifolds. The third project is a program to prove a conjecturewhich asserts the existence of periodic orbits on all level setsnear a nondegenerate symplectic critical submanifold of aHamiltonian. This is a generalization of some similar conjecturesof Arnold which concern periodic orbits of a charged particlemoving in a magnetic field. The first step is to construct aFloer-type invariant for the underlying variationalprinciple. Once it is rigorously defined, this should quicklylead to many new existence results. It is also hoped that thisinvariant can be used to augment Symplectic Field Theory byallowing one to split a symplectic manifold along certainhypersurfaces which are not of contact type.Hamiltonian flows are used to model many important physicalsystems in which energy is conserved. Such systems includeplanets and satellites moving under their mutual gravitationalattraction, a charged particle moving in an electro-magneticfield, and the flow of an incompressible ideal fluid. Thesemotions are often quite complex and one way to begin tounderstand their global behavior is to look for repeatingpatterns, i.e., periodic orbits. While most Hamiltonian flowshave many periodic orbits, it is usually a difficult problem toestablish their existence at a fixed energy level. This problemis a central theme in the study of Hamiltonian flows and, inmodern times, has been shown to be deeply related to the shape ofthe space on which the flow is defined. The projects in thisproposal study various aspects of this relation. In the first twoprojects we use Hamiltonian flows to define and computesymplectic invariants. The last project involves the constructionof a new symplectic invariant which should lead to new existenceresults for periodic orbits of Hamiltonian flows which describethe motion of a charged particle in a magnetic field.