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Review of The Structure and Stability of Persistence Modules by Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot

The Structure and Stability of Persistence Modules provides a comprehensive introduction to the theory of persistence in applied algebraic topology. The book is divided into six parts: Introduction, Persistence Modules, Rectangle Measures, Interleaving, The Isometry Theorem, and Variations. After providing a brief history and overview of the field, the authors dive into the definition of a persistence module and discuss its relations to persistence diagrams and notions of distances between persistence modules throughout the book. The target audience for this book is both newcomers and experts alike, providing both a foundation for newcomers to build off of and a general framework for experts. In particular, we believe this book is most accessible to graduate students in mathematics, who already have some familiarity with the idea of persistence. Overall, the authors capture the attention of the audience through the abundance of motivating questions, detailed figures, and hints towards applications of the theory, while maintaining the necessary rigor within definitions and proofs.