Persistent homology for functionals

TL;DR

This paper establishes topological conditions under which persistent homology modules of certain functionals admit persistence diagrams, enabling a modern proof of the Unstable Minimal Surface Theorem and connecting topology with variational analysis.

Contribution

It introduces topological conditions ensuring persistence diagrams for a broad class of functionals, extending the applicability of persistent homology in variational problems.

Findings

Persistence diagrams exist for the class of functionals satisfying the conditions.

Generalized Morse inequalities hold for these functionals.

Reformulation of the Unstable Minimal Surface Theorem proof using persistent homology.

Abstract

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework.