Moscow Mathematical Journal

Volume 17, Issue 4, October–December 2017 pp. 757–786.

Persistence Modules with Operators in Morse and Floer Theory

Authors: Leonid Polterovich (1), Egor Shelukhin (2), and Vukašin Stojisavljević (1)
Author institution:(1) School of Mathematical Sciences, Tel Aviv University
(2) IAS, Princeton, and DMS at U. of Montreal

Summary:

We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along with a few other geometric situations. We provide sample applications to the C⁰-geometry of Morse functions and to Hofer’s geometry of Hamiltonian diffeomorphisms that go beyond spectral invariants and traditional persistent homology.

2010 Math. Subj. Class. Primary: 53D40; Secondary: 58E05.

Keywords: Symplectic manifold, Hamiltonian diffeomorphism, Floer homology, persistence module, barcode.

Contents Full-Text PDF