Persistence module and Schubert calculus

TL;DR

This paper introduces a novel multiplication on persistence diagrams using Schubert calculus, linking algebraic geometry and topological data analysis through the structure of persistence modules and their Schubert decomposition.

Contribution

It establishes a new algebraic operation on persistence diagrams based on Schubert calculus, connecting topological persistence with algebraic geometry.

Findings

Defines a multiplication on persistence diagrams via Schubert calculus.

Shows isomorphism classes of persistence modules correspond to Schubert cells.

Provides an interpretation of the multiplication as intersections of algebraic varieties.

Abstract

A multiplication on persistence diagrams is introduced by means of Schubert calculus. The key observation behind this multiplication comes from the fact that the representation space of persistence modules has the structure of the Schubert decomposition of a flag. In particular, isomorphism classes of persistence modules correspond to Schubert cells, thereby the Schubert calculus naturally defines a multiplication on persistence diagrams. The meaning of the multiplication on persistence diagrams is carried over from that on Schubert calculus, i.e., algebro-geometric intersections of varieties of persistence modules.