An Algebraic Introduction to Persistence

TL;DR

This paper provides an algebraic perspective on persistence, focusing on the representation theory of posets and their applications in topology, geometry, and data analysis.

Contribution

It introduces the algebraic foundations of persistence using poset representations and surveys key results, applications, and open problems in the field.

Findings

Persistence representations admit a metric structure via interleaving distance.

Algebraic properties of poset representations are crucial for understanding stability.

Applications span topological data analysis and geometric contexts.

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.