The Structure and Stability of Persistence Modules

This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such modules, providing a sound mathematical framework for the study of persistence diagrams. Completely self-contained, this brief introduces the notion of persistence measure and makes extensive use of a new calculus of quiver representations to facilitate explicit computations.
Appealing to both beginners and experts in the subject, The Structure and Stability of Persistence Modules provides a purely algebraic presentation of persistence, and thus complements the existing literature, which focuses mainly on topological and algorithmic aspects.

Provides a comprehensive treatment of 1-parameter persistence modules Presents new tools for studying persistence modules in wide generality Introduces the measure-theory approach to persistence diagrams Offers a definitive treatment of the stability theorem

Zusammenfassung

"This book is a very nice contribution to the subject of Topological Data Analysis. In this slim volume, the novice will find a collection of main results with their proofs and many references; additionally, experts will see persistence developed more generally than usual using measure theory. ... There are many synthesizing comments throughout the text to help the reader put the material in context, and the writing itself is lucid." (Michele Intermont, MAA Reviews, October, 2017)

"This monograph develops the theory of persistence modules over the real line in a manner that is well-motivated, accessible, thorough, and self-contained." (Henry Hugh Adams, Mathematical Reviews, October, 2017)

"This book offers an excellent introduction to anyone interested in understanding the fundamentals of persistent homology. The exposition is clear, concise and easy to read. ... A fair overview of similar results appearing elsewhere is given, and an extensive list of suggested further reading is provided for the inspired reader. ... In short, the book offers a self-contained introduction to topics such as persistence modules, persistence diagrams, interleavings, and the famous algebraic stability theorem." (Magnus Bakke Botnan, zbMATH, 2017)

"This monograph develops the theory of persistence modules over the real line in a manner that is well-motivated, accessible, thorough, and self-contained. ... In this monograph, the theory of persistence modules over the reals is presented from scratch, with the main results and their proofs in a natural framework that is convenient to learn and to use." (Henry Hugh Aams, Mathematical Reviews, 2017)

Inhalt
Introduction.- Persistence Modules.- Rectangle Measures.- Interleaving.- The Isometry Theorem.- Variations.- References.- Index.