Floer Homology and Closed Orbits of Hamiltonian Systems

项目负责人：田刚，刘刚。本项目的目的是继续研究刘刚和田刚在花同调和辛流形上哈密顿系统动力学中的相对虚模环构造的应用。利用这一构造，研究人员将半正情形下的花同调推广到所有闭辛流形，从而完全解决了非简并阿诺德猜想。通过对这一构造的某些改进，他们建立了某些gw不变量的不消失与哈密顿系统闭轨道的存在之间的一般关系。作为其应用之一，他们解决了一个稳定版本的温斯坦猜想。在此基础上，主要研究人员提出进一步研究简并情形下的Arnold猜想和Weinstein猜想中一些未解决的问题。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，作为控制这些系统中运动的基本方程。哈密顿系统的动力学描述了“经典”世界的演化。理解哈密顿系统动力学的一个重要步骤是理解它们最简单的动力学行为，即周期轨道。这里的基本问题，被称为阿诺德猜想和温斯坦猜想，是关于哈密顿系统的闭合轨道的存在和数量。这两个猜想一直被认为是辛拓扑学科的主要指导问题。本课题研究人员提出的求解非简并Arnold猜想和稳定Weinstein猜想的方法，为研究简并情形下的Arnold猜想和一般辛流形的Weinstein猜想打开了大门。

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.