Filtered Floer Theory and Hamiltonian Dynamics

项目申请：DMS 1105700，项目负责人：Michael usher。本项目将使用多种方法，主要来源于Floer理论，研究辛流形的哈密顿微分同态问题。该项目的目标包括：扩展已知哈密顿微分同构群（或其全称覆盖）承认Calabi拟态的流形类；研究关于哈密顿微分同态群和拉格朗日子流形空间上的Hofer度量的全局几何问题（例如，这些空间何时具有无限直径？）并发展了一个新的相对辛能力家族，这些能力是通过以一种新颖的方式使用弗洛尔理论构建的。为了实现这一目标，PI将利用来自flower复合物的自然实值过滤的工具，包括完善的Oh-Schwarz光谱不变量和一个新的不变量，边界深度，这是由首席研究员在2009年首次引入的。该项目的一个辅助目标是更好地理解这两个有用的不变量的行为。辛流形的哈密顿微分同态可以用来对能量守恒的经典物理系统进行数学建模。因此，从行星和卫星的运动（可能应用于成本较低的空间任务规划）到湍流的流动，它们在研究范围非常广泛的现象方面都具有相关性。20多年前，Hofer在辛流形的哈密顿微分同态群上发现了一个显著的几何结构；尽管付出了很多努力，但这种几何的一些基本方面仍然知之甚少。这个项目的大部分目的是获得关于Hofer几何的新结果，并澄清辛流形的哈密顿微分同态的行为如何与流形的其他性质相关。

AbstractAward: DMS 1105700, Principal Investigator: Michael UsherThis project will use a variety of methods, mostly arising from Floer theory, to study questions relating to Hamiltonian diffeomorphisms of symplectic manifolds. The goals of the project include: expanding the class of manifolds for which the Hamiltonian diffeomorphism group (or its universal cover) is known to admit Calabi quasi-morphisms; investigating global geometric questions about Hofer's metrics on the Hamiltonian diffeomorphism group and on spaces of Lagrangian submanifolds (for instance, when do these spaces have infinite diameter?); and developing a new family of relative symplectic capacities that are constructed by using Floer theory in a novel way. To achieve this, the PI will make use of tools coming from the natural real-valued filtrations on Floer complexes, including the well-established Oh-Schwarz spectral invariants and also a newer invariant, the boundary depth, that was first introduced by the principal investigator in 2009. An auxiliary goal of the project is to obtain a better understanding of the behavior of these two useful invariants.Hamiltonian diffeomorphisms of symplectic manifolds can be used to mathematically model those classical physical systems in which energy is conserved. Thus they are relevant in the study of a very wide range of phenomena, from the motion of planets and satellites (with potential applications to lower-cost space mission planning) to the flow of turbulent fluids. A remarkable geometric structure on the group of Hamiltonian diffeomorphisms of a symplectic manifold was discovered by Hofer over 20 years ago; despite much effort some basic aspects of this geometry remain poorly understood. Much of this project is aimed at obtaining new results about Hofer's geometry, and at clarifying how the behavior of Hamiltonian diffeomorphisms of a symplectic manifold is related to other properties of the manifold.