Floer Homology and Closed Orbits of Hamiltonian Systems

摘要 提案：DMS 9802460主要研究者：田刚、刘刚 这个项目的目的是继续调查刘和 论相对虚模构造的应用 Floer同调中的循环和上的Hamilton系统的动力学 辛流形 通过使用这种结构，主要 研究人员能够将Floer同源性从半阳性 的情况下，所有封闭辛流形，从而解决 非退化Arnold猜想 通过使用某些改进 在这一结构中，他们建立了一种普遍的关系， 某些GW-不变量的非零性和闭轨的存在性 Hamilton系统的。 作为应用之一，他们解决了一个 温斯坦猜想的稳定化版本 基于这些结果， 主要调查人员建议对阿诺德进行进一步调查 猜想退化的情况和其他一些悬而未决的情况下， 温斯坦猜想。 哈密顿方程起源于经典力学，天体力学 和许多其他物理系统作为基本方程， 在这样的系统中。 哈密顿系统的动力学描述了 “古典”世界的演变。 的重要步骤之一 要理解哈密顿系统的动力学， 最简单的动力学行为，周期轨道。 根本问题 在这里，被称为阿诺德猜想和温斯坦猜想，是 关于哈密顿系统的闭轨的存在性和个数。 这两个问题已被认为是主要的指导问题， 辛拓扑学的主题。 开发的方法 该项目的研究人员解决了非退化阿诺德猜想 并稳定了温斯坦猜想， 退化情形的Arnold猜想和Weinstein猜想 对于一般辛流形。

Abstract Proposal: DMS 9802460 Principal Investigators: Gang Tian and Gang Liu The purpose of this project is to continue the investigation of Liu and Tian on the applications of their construction of relative virtual moduli cycles in Floer homology and the dynamics of Hamiltonian systems on symplectic manifolds. By using this construction, the principal investigators were able to extend Floer homology from the semi-positive case to all closed symplectic manifolds and consequently to solve the non-degenerate Arnold conjecture completely. By using certain refinements of this construction, they established a general relationship between non-vanishing of certain GW-invariants and the existence of closed orbits of Hamiltonian systems. As one of the applications of this, they solved a stabilized version of Weinstein conjecture. Base on these results, the principal investigators propose further investigations on the Arnold conjecture for degenerate case and some other unsettled cases of the Weinstein conjecture. Hamiltonian equations arise from classical mechanics, celestial mechanics and many other physical systems as fundamental equations governing the motions in such systems. The dynamics of Hamiltonian systems describes the evolution of the "classical" world. One of the important steps to understand the dynamics of Hamiltonian systems is to understand their simplest dynamic behavior, the periodical orbits. The basic questions here, known as the Arnold conjecture and the Weinstein conjecture, are about the existence and the number of closed orbits of Hamiltonian systems. Both of these conjectures have been considered as main guiding problems in the subject of symplectic topology. The methods developed by the investigators of this project to solve the non-degenerate Arnold conjecture and stabilized Weinstein conjecture have opened the door for investigating the Arnold conjecture for the degenerate case and the Weinstein conjectur e for general symplectic manifolds.