Hamiltonian dynamic and persistent homology

It has been recently noticed that persistent homology, originated from topological data analysis, can be used in the study of symplectic topology, especially in Floer-type theory and Hofer geometry. One of its applications is on the question how far away a Hamiltonian diffeomorphism from being autonomous (this question is studied by Polterovich-Shelukhin in the symplectically aspherical case). In this talk, I will first explain the language of persistent homology and its extended version involving Floer-Novikov theory (which is my joint work with Mike Usher). Moreover, I will demonstrate how to use persistent homology to construct symplectic invariant so that we can give a quantitative answer on the question above. My main result generalizes the main theorem from Polterovich-Shelukhin’s paper on this topic. Moreover, the algebraic framework formulated in the process of answering this question serves as a promising and useful tool to answer possibly more Hamiltonian dynamic questions.