Sketches of a platypus: persistent homology and its algebraic foundations

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.

💡 Research Summary

The paper “Sketches of a platypus: persistent homology and its algebraic foundations” is a comprehensive survey that examines the various algebraic frameworks used to formalize persistent homology and evaluates how each framework influences the kinds of results that can be proved. The authors begin by recalling the impact of persistent homology on applied fields such as data analysis, signal processing, and biology, emphasizing that the method provides topological summaries (barcodes and diagrams) that capture shape information invisible to classical statistics.

Four principal algebraic perspectives are identified. The first is the classical module‑theoretic approach, where a persistence module is a graded module over the polynomial ring 𝕜