Sketches of a platypus: persistent homology and its algebraic foundations

TL;DR

This survey explores the algebraic foundations of persistent homology, examining different theoretical choices, their implications, and future research directions in applying algebraic topology to data analysis.

Contribution

It provides a comprehensive overview of the various algebraic frameworks used in persistent homology and discusses their differences and potential research paths.

Findings

Different algebraic foundations influence the results in persistent homology.

The survey highlights key theoretical choices and their impact on applications.

Future research directions are proposed to unify or improve foundational approaches.

Abstract

The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are rooted in choosing a particular algebraic foundational theory to describe persistent homology, and applying results from that theory to prove useful and important results. In this survey paper, we shall examine the various choices in use, and what they allow us to prove. We shall also discuss the inherent differences between the choices people use, and speculate on potential directions of research to resolve these differences.