Preface; Acknowledgments; Contents; 1 Introduction; 1.1 A Brief History of Persistence; 1.2 Main Contributions; 1.3 Application: Stable Descriptors for Metric Spaces; 1.4 Application: Stable Clustering Using Persistence; 1.5 Recommended Reading; 1.6 Organisation; 1.7 Multisets; 2 Persistence Modules; 2.1 Persistence Modules Over a Real Parameter; 2.2 Index Posets; 2.3 Module Categories; 2.4 Interval Modules; 2.5 Interval Decomposition; 2.6 The Decomposition Persistence Diagram; 2.7 Quiver Calculations; 3 Rectangle Measures; 3.1 The Persistence Measure; 3.2 The Persistence Measure (Continued)

3.3 Abstract r-Measures3.4 Equivalence of Measures and Diagrams; 3.5 Non-finite Measures; 3.6 Measures and Diagrams in the Extended Plane; 3.7 The Measure Persistence Diagram; 3.8 Tameness; 3.9 Tameness (Continued); 3.10 Vanishing Lemmas; 3.11 Vanishing Lemmas (Continued); 3.12 Finite Approximations; 4 Interleaving; 4.1 Shifted Homomorphisms; 4.2 Interleaving; 4.3 Interleaving (Continued); 4.4 The Interpolation Lemma; 4.5 The Interpolation Lemma (Continued); 5 The Isometry Theorem; 5.1 The Interleaving Distance; 5.2 The Bottleneck Distance; 5.3 The Bottleneck Distance (Continued)

5.4 The Isometry Theorem5.5 The Converse Stability Theorem; 5.6 The Stability Theorem; 5.7 The Measure Stability Theorem; 5.8 The Measure Stability Theorem (Continued); 6 Variations; 6.1 Partial Interleavings; 6.2 Extended Persistence; References; Index