Persistence modules with operators in Morse and Floer theory

By L. Polterovich, E. Shelukhin, Vukavsin Stojisavljevi'c. 2017

47 citations recorded by the source corpus.

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 $C^0$-geometry of Morse functions and to Hofer's geometry of Hamiltonian diffeomorphisms, that go beyond spectral invariants and traditional persistent homology.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6775955f403e3c7aec5c1614cc2a4fdcd151a4c3

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In Computational Theory and Mathematics

1. E. Shelukhin
2. A. Kislev

In Geometry and Topology

1. E. Shelukhin
2. A. Kislev

In Mathematical Physics

1. E. Shelukhin
2. A. Kislev

Citations in the corpus

• Bounds on spectral norms and barcodes (2018, background): “Finally, as in [93, 94], continuation maps corresponding to changing the J component of the perturbation datum induce isomorphisms of persistence modules, hence when (L,H) is non-degenerate, the Hamiltonian term in D can be taken to be identically zero, and we have a well-defined persistence modules”