# Persistence modules with operators in Morse and Floer theory

## TL;DR

This paper introduces persistence modules with operators to incorporate additional geometric structures in Morse and Floer theory, enabling new insights into symplectic geometry and Hamiltonian dynamics beyond traditional methods.

## Contribution

It defines a novel framework for persistence modules with operators, capturing extra structures in Floer-type modules and applying them to symplectic and Morse geometric contexts.

## Key findings

- Enhanced understanding of $C^0$-geometry of Morse functions
- Applications to Hofer's geometry of Hamiltonian diffeomorphisms
- Beyond spectral invariants and traditional persistent homology

## Abstract

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.