Filtered Floer Theory and Hamiltonian Dynamics

过滤弗洛尔理论和哈密顿动力学

• 批准号: 1105700
• 负责人: Michael Usher
• 金额: $ 10.06万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105700&HistoricalAwards=false
• 关键词: Filtered; Floer; Theory; Hamiltonian; Dynamics

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.

AbstractAward：DMS 1105700，首席研究员：Michael Usher这个项目将使用各种方法，主要是从Floer理论中产生的，来研究与辛流形的Hamilton同构有关的问题。 该项目的目标包括：扩展已知Hamilton同构群（或其泛覆盖）的流形类，使其允许Calabi拟态射;研究关于霍费尔在Hamilton同构群和拉格朗日子流形空间上的度量的全局几何问题（例如，这些空间何时具有无穷大直径？）;并利用Floer理论以一种新颖的方式构造了一族新的相对辛容度。为了实现这一目标，PI将利用来自Floer复形上的自然实值过滤的工具，包括完善的Oh-Schwarz谱不变量和一个较新的不变量，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Hamilton拓扑同构可以用来对能量守恒的经典物理系统进行数学建模。 因此，它们与研究范围非常广泛的现象有关，从行星和卫星的运动（可能应用于低成本空间使命规划）到湍流流体的流动。一个显着的几何结构组的哈密顿同态的辛流形被发现的霍费尔超过20年前，尽管许多努力的一些基本方面，这种几何仍然知之甚少。 这个项目的大部分是为了获得新的结果有关霍费尔的几何，并在澄清如何行为的哈密顿同态的辛流形是相关的其他性质的流形。