Symplectic Floer Theory and Persistent Homology

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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.