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.

摘要奖项：DMS 1105700，首席研究员：Michael Usher 该项目将使用多种方法（主要来自 Floer 理论）来研究与辛流形哈密顿微分同胚相关的问题。 该项目的目标包括：扩展已知承认卡拉比拟态射的哈密顿微分同胚群（或其通用覆盖）的流形类；研究关于哈密顿微分同胚群和拉格朗日子流形空间的 Hofer 度量的全局几何问题（例如，这些空间何时具有无限直径？）；并开发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 Floer 复合体的自然实值过滤的工具，包括完善的 Oh-Schwarz 谱不变量以及一个较新的不变量，即边界深度，它是由校长首先引入的2009 年的研究员。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。辛流形的哈密尔顿微分同胚可以在数学上用于对能量守恒的经典物理系统进行建模。 因此，它们与广泛现象的研究相关，从行星和卫星的运动（具有低成本空间任务规划的潜在应用）到湍流流体的流动。 Hofer 在 20 多年前发现了辛流形哈密顿微分同胚群上的一个显着的几何结构；尽管付出了很多努力，但这种几何学的一些基本方面仍然知之甚少。 该项目的大部分目的是获得有关 Hofer 几何的新结果，并阐明辛流形的哈密顿微分同胚的行为如何与流形的其他属性相关。