Topological Persistence in Geometry and Analysis

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https://sites.google.com/site/polterov/miscellaneoustexts/topological-persistence-in-geometry-and-analysis Preface I A primer of persistence modules Definition and first examples Persistence modules Morphisms Interleaving distance First example: interval modules Morse persistence modules and approximation Rips modules and the Gromov-Hausdorff distance Barcodes The Normal Form Theorem Bottleneck distance and the Isometry theorem Corollary: stability theorems Persistence modules of locally finite type Proof of the Isometry theorem An outline Matchings for surjections and injections Monotonicity with respect to injections and surjections Induced matchings construction Main lemmas and proof of the theorem Proofs of Lemma 3.3.1 and Lemma 3.3.2 Proof of Lemma 3.3.1. Proof of Lemma 3.3.2. What can we read from a barcode? Infinite bars and characteristic exponents Characteristic exponents Boundary depth and approximation The multiplicity function Representations on persistence modules Theoretical development Applications in geometry II Applications to metric geometry and function theory Applications of Rips complexes delta - hyperbolic spaces Cech complex, Rips complex and topological data analysis Manifold Learning Topological function theory Prologue Invariants of upper triangular matrices Simplex counting method A combinatorial lemma Bars and oscillation The length of the barcode The fundamental inequality The Banach indicatrix Normal lifts of the level lines Approximation by trigonometric polynomials III Persistent homology in symplectic geometry A concise introduction to symplectic geometry Hamiltonian dynamics Symplectic structures on manifolds Group of Hamiltonian diffeomorphisms Hofer's bi-invariant geometry A short tour in coarse geometry Zoo of symplectic embeddings Hamiltonian persistence modules Conley-Zehnder index Filtered Hamiltonian Floer theory Constraints on full powers Non-contractible class version Barcodes for Hamiltonian homeomorphisms Symplectic persistence modules Liouville manifolds Symplectic persistence module Examples of SH(U) Symplectic Banach-Mazur distance Functorial properties Applications Computations Bibliography