Topological Persistence in Geometry and Analysis (University Lecture Series)

The theory of persistence modules originated in topological data analysis and became an active area of research in algebraic topology. This book provides a concise and self-contained introduction to persistence modules and focuses on their interactions with pure mathematics, bringing the reader to the cutting edge of current research. In particular, the authors present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, they discuss topological function theory, which provides new insight into oscillation of functions. The book is accessible to readers with a basic background in algebraic and differential topology. Cover Title page Preface Part 1 . A primer of persistence modules Chapter 1. Definition and first examples 1.1. Persistence modules 1.2. Morphisms 1.3. Interleaving distance 1.4. Morse persistence modules and approximation 1.5. Rips modules and the Gromov-Hausdorff distance Chapter 2. Barcodes 2.1. The Normal Form Theorem 2.2. Bottleneck distance and the Isometry Theorem 2.3. Corollary: Stability theorems 2.4. Persistence modules of locally finite type Chapter 3. Proof of the Isometry Theorem 3.1. An outline 3.2. Matchings for surjections and injections 3.3. Main lemmas and proof of the theorem 3.4. Proofs of Lemma 3.3.1 and Lemma 3.3.2 Chapter 4. What can we read from a barcode? 4.1. Infinite bars and characteristic exponents 4.2. Boundary depth and approximation 4.3. The multiplicity function 4.4. Representations on persistence modules Part 2 . Applications to metric geometry and function theory Chapter 5. Applications of Rips complexes 5.1. δ - hyperbolic spaces 5.2. Čech complex, Rips complex and topological data analysis 5.3. Manifold learning Chapter 6. Topological function theory 6.1. Prologue 6.2. Invariants of upper triangular matrices 6.3. Simplex counting method 6.4. The length of the barcode 6.5. Approximation by trigonometric polynomials Part 3 . Persistent homology in symplectic geometry Chapter 7. A concise introduction to symplectic geometry 7.1. Hamiltonian dynamics 7.2. Symplectic structures on manifolds 7.3. The group of Hamiltonian diffeomorphisms 7.4. Hofer’s bi-invariant geometry 7.5. A short tour in coarse geometry 7.6. Zoo of symplectic embeddings Chapter 8. Hamiltonian persistence modules 8.1. Conley-Zehnder index 8.2. Filtered Hamiltonian Floer theory 8.3. Constraints on full powers 8.4. Non-contractible closed orbits 8.5. Barcodes for Hamiltonian homeomorphisms Chapter 9. Symplectic persistence modules 9.1. Liouville manifolds 9.2. Symplectic persistence modules 9.3. Examples of SH_{*}(U) 9.4. Symplectic Banach-Mazur distance 9.5. Functorial properties 9.6. Applications 9.7. Computations Bibliography Notation Index Subject Index Name Index Back Cover