Symplectic Floer Theory and Persistent Homology

辛弗洛尔理论和持久同调

• 批准号: 1509213
• 负责人: Michael Usher
• 金额: $ 17.58万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1509213&HistoricalAwards=false
• 关键词: Symplectic; Floer; Theory; Persistent; Homology

This project aims to synthesize certain ideas coming from two rather distinct subjects in geometry and topology: persistent homology, which was created as a tool for studying the topological structure of data sets; and symplectic Floer theory, which concerns certain properties of the geometric transformations ("Hamiltonian diffeomorphisms") that underlie classical mechanics. Despite their different origins, it has recently been appreciated that persistent homology and symplectic Floer theory share key algebraic structures, and the time is ripe to exploit these parallels and apply the insights gained during the separate developments of these two subjects over the last several years in order to learn more about each of them. In particular, methods from persistent homology will make it possible to prove new results about a natural geometry on the group of Hamiltonian diffeomorphisms and about the relationships between the fixed points of different Hamiltonian diffeomorphisms. The starting point for this work will be recent results that adapted the construction of "barcodes" from persistent homology to the context of Floer theory over Novikov fields by using a novel approach involving non-Archimedean singular value decompositions, and proved a version of the Bottleneck Stability Theorem for these new barcodes. Building on this algebraic foundation, the investigator expects to express and generalize the notion of extended persistence in a way that allows one to streamline arguments involving action windows in Floer theory; among other things this may lead to new proofs of Conley conjecture-type results and generalizations of recent work on autonomous Hamiltonians. The project will also study the properties of certain symplectic capacities built from filtered Floer-theoretic invariants, leading to new relations between lower bounds for the Hofer norm and the properties of periodic orbits of Hamiltonian systems.

该项目旨在综合来自几何和拓扑学中两个截然不同的学科的某些想法：持久同源性，它是作为研究数据集拓扑结构的工具而创建的；辛弗洛尔理论，它涉及经典力学基础的几何变换（“哈密尔顿微分同胚”）的某些性质。尽管它们的起源不同，但最近人们认识到，持久同调和辛弗洛尔理论共享关键的代数结构，利用这些相似之处并应用在过去几年中这两个学科单独发展中获得的见解的时机已经成熟。以便更多地了解他们每个人。特别是，来自持久同调的方法将使得证明关于哈密顿微分同胚群上的自然几何以及关于不同哈密顿微分同胚的不动点之间的关系的新结果成为可能。这项工作的起点将是最近的结果，通过使用涉及非阿基米德奇异值分解的新颖方法，将“条形码”的构造从持久同源性适应到诺维科夫场上的弗洛尔理论的背景，并证明了瓶颈的一个版本这些新条形码的稳定性定理。在此代数基础上，研究者期望以一种能够简化涉及 Floer 理论中的动作窗口的论证的方式来表达和概括扩展持久性的概念。除其他外，这可能会导致康利猜想类型结果的新证明以及自治哈密顿量最近工作的概括。该项目还将研究由过滤弗洛尔理论不变量构建的某些辛能力的性质，从而得出霍弗范数下界与哈密顿系统周期轨道性质之间的新关系。