An Exposition on the Algebra and Computation of Persistent Homology

TL;DR

This paper explores the algebraic foundations of persistent homology, connecting classical matrix reduction methods with modern functor-based perspectives to deepen understanding of computational topology.

Contribution

It bridges traditional matrix algorithms with the modern functorial approach to persistence modules, clarifying their algebraic relationship.

Findings

Unified algebraic framework for persistent homology

Enhanced understanding of matrix reduction in topological data analysis

Connection between classical and modern persistence theories

Abstract

We discuss the algebra behind the matrix reduction algorithm for persistent homology, as presented in the paper ''Computing Persistent Homology'' by Afra Zomorodian and Gunnar Carlsson, in the lens of the more modern characterization of persistence modules as functors from a poset category to a category of vector spaces over a field adopted by authors such as Peter Bubenik, Frederik Chazal, and Ulrich Bauer.