Persistent Homology and Equivariance in Data Analysis: A Topological Introduction

Abstract

This new book is intended as a first elementary introduction to Topological Data Analysis for mathematics students seeking a rigorous account of the foundations of persistent homology, as well as for computer scientists interested in its theoretical underpinnings. The exposition is as self-contained as possible: all the required background is recalled when needed, and only a few standard results are cited without proof. One section of the book, devoted to monodromy in biparameter persistence (Section 4.4), requires more advanced knowledge of algebraic topology. Persistent homology can be introduced from different perspectives, reflecting the variety of mathematical languages that have shaped its development over the years. Some approaches emphasize the algebraic foundations of the theory, while others highlight its topological essence. In this book, we adopt the latter viewpoint - the one that historically marked the birth of the subject - because we believe it offers both conceptual clarity and pedagogical effectiveness, making it particularly suitable for undergraduate and early graduate students. This book differs from existing introductory texts in several respects. First, it adopts a functional viewpoint: rather than representing data as finite (pseudo-)metric spaces, it treats them as functions encoding the information to be analyzed. This interpretative framework allows data to be viewed as measurable objects and highlights the role of observers and their equivariances in the analysis process. Second, this perspective provides a natural bridge between Topological Data Analysis and machine learning through the theory of Group Equivariant Non-Expansive Operators (GENEOs), which offers a mathematically grounded framework for incorporating symmetries and invariances into learning systems.