Symplectic Floer Theory and Persistent Homology

该项目旨在综合来自几何学和拓扑学中两个截然不同的学科的某些思想：持久同调，它是作为研究数据集拓扑结构的工具而创建的；以及辛花理论，它涉及经典力学基础上的几何变换的某些性质（“哈密顿微分同态”）。尽管它们的起源不同，但最近人们认识到，持久同调和辛花理论共享关键的代数结构，并且利用这些相似之处的时机已经成熟，并应用在过去几年这两个主题的单独发展中获得的见解，以便更多地了解它们。特别是，用持久同调的方法可以证明关于哈密顿微分同态群的自然几何和不同哈密顿微分同态不动点之间关系的新结果。这项工作的起点将是最近的结果，通过使用一种涉及非阿基米德奇异值分解的新方法，将“条形码”的构造从持久同调调整到Novikov场上的Floer理论的背景下，并证明了这些新条形码的瓶颈稳定性定理的一个版本。建立在这个代数基础上，研究者期望以一种允许人们在Floer理论中简化涉及动作窗口的论证的方式来表达和概括扩展持久性的概念；除其他事项外，这可能会导致康利猜想类型结果的新证明和最近关于自治哈密顿量的工作的推广。该项目还将研究从过滤花理论不变量建立的某些辛容量的性质，从而导致Hofer范数下界与哈密顿系统周期轨道性质之间的新关系。

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.