Frédéric Chazal & Vin de Silva 
The Structure and Stability of Persistence Modules [PDF ebook] 

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.

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