Filtered Floer Theory and Hamiltonian Dynamics

摘要奖：DMS 1105700，首席研究员：Michael Usher这个项目将使用主要源于Floer理论的各种方法来研究与辛流形的哈密尔顿微分同态有关的问题。该项目的目标包括：扩展已知哈密尔顿微分同胚群(或其泛覆盖)允许Calabi拟同态的流形类别；研究哈密尔顿微分同胚群和拉格朗日子流形空间(例如，这些空间何时有无限直径？)上的Hofer度量的全局几何问题；以及以一种新颖的方式利用Floer理论构造一族新的相对辛容。为了实现这一点，PI将利用来自Floer复形上的自然实值过滤的工具，包括著名的oh-Schwarz谱不变量以及由首席研究员于2009年首次引入的较新的不变量-边界深度。这个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那些能量守恒的经典物理系统进行数学建模。因此，它们与研究非常广泛的现象有关，从行星和卫星的运动(可能应用于低成本的空间飞行任务规划)到湍流流体的流动。20多年前，Hofer在辛流形的哈密尔顿微分同态群上发现了一种显着的几何结构；尽管做了很多工作，但对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胚行为如何与该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.