An Algebraic Introduction to Persistence

2026-04-08

AI Summary

This study investigates the algebraic stability of linear representations of posets under perturbations, with a focus on the deep interplay between the metric structure induced by the interleaving distance in persistent homology and underlying algebraic properties. By embedding persistence theory within a unified categorical framework of poset representation theory, the work systematically uncovers fundamental algebraic structures and stability mechanisms. It integrates tools from homological algebra, category theory, and topological data analysis, clarifying connections between this theoretical framework and geometric as well as applied mathematical contexts. The paper also identifies several key challenges and open problems, offering a fresh perspective on the algebraic foundations of persistent homology.

Abstract

We introduce persistence with an emphasis on its algebraic foundations, using the representation theory of posets. Linear representations of posets arise in several areas of mathematics, including the representation theory of quivers and finite dimensional algebras, Morse theory and other areas of geometry, as well as topological inference and topological data analysis -- often via persistent homology. In some of these contexts, the category of poset representations of interest admits a metric structure given by the so-called interleaving distance. Persistence studies the algebraic properties of these poset representations and their behavior under perturbations in the interleaving distance. We survey fundamental results in the area and applications to pure and applied mathematics, as well as theoretical challenges and open questions.

Problem

• persistence
• poset representations
• interleaving distance
• persistent homology
• representation theory

Innovation

• persistence
• poset representations
• interleaving distance
• representation theory
• persistent homology