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E-raamat: Topological Persistence in Geometry and Analysis

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  • Formaat: 128 pages
  • Sari: University Lecture Series
  • Ilmumisaeg: 30-Jun-2020
  • Kirjastus: American Mathematical Society
  • ISBN-13: 9781470456795
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Topological Persistence in Geometry and AnalysisSuurem pilt
  • Formaat: 128 pages
  • Sari: University Lecture Series
  • Ilmumisaeg: 30-Jun-2020
  • Kirjastus: American Mathematical Society
  • ISBN-13: 9781470456795
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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.
Preface vii
Part 1 A primer of persistence modules
1(46)
Chapter 1 Definition and first examples
3(10)
1.1 Persistence modules
3(2)
1.2 Morphisms
5(2)
1.3 Interleaving distance
7(2)
1.4 Morse persistence modules and approximation
9(1)
1.5 Rips modules and the Gromov-Hausdorff distance
9(4)
Chapter 2 Barcodes
13(10)
2.1 The Normal Form Theorem
13(6)
2.2 Bottleneck distance and the Isometry Theorem
19(1)
2.3 Corollary: Stability theorems
20(1)
2.4 Persistence modules of locally finite type
20(3)
Chapter 3 Proof of the Isometry Theorem
23(10)
3.1 An outline
23(1)
3.2 Matchings for surjections and injections
24(5)
3.3 Main lemmas and proof of the theorem
29(1)
3.4 Proofs of Lemma 3.3.1 and Lemma 3.3.2
30(3)
Chapter 4 What can we read from a barcode?
33(14)
4.1 Infinite bars and characteristic exponents
33(3)
4.2 Boundary depth and approximation
36(3)
4.3 The multiplicity function
39(2)
4.4 Representations on persistence modules
41(6)
Part 2 Applications to metric geometry and function theory
47(28)
Chapter 5 Applications of Rips complexes
49(12)
5.1 δ - hyperbolic spaces
49(5)
5.2 Cech complex, Rips complex and topological data analysis
54(3)
5.3 Manifold learning
57(4)
Chapter 6 Topological function theory
61(14)
6.1 Prologue
61(2)
6.2 Invariants of upper triangular matrices
63(3)
6.3 Simplex counting method
66(2)
6.4 The length of the barcode
68(3)
6.5 Approximation by trigonometric polynomials
71(4)
Part 3 Persistent homology in symplectic geometry
75(40)
Chapter 7 A concise introduction to symplectic geometry
77(10)
7.1 Hamiltonian dynamics
77(1)
7.2 Symplectic structures on manifolds
78(1)
7.3 The group of Hamiltonian diffeomorphisms
79(1)
7.4 Hofer's bi-invariant geometry
80(2)
7.5 A short tour in coarse geometry
82(1)
7.6 Zoo of symplectic embeddings
83(4)
Chapter 8 Hamiltonian persistence modules
87(12)
8.1 Conley-Zehnder index
87(2)
8.2 Filtered Hamiltonian Floer theory
89(5)
8.3 Constraints on full powers
94(2)
8.4 Non-contractible closed orbits
96(2)
8.5 Barcodes for Hamiltonian homeomorphisms
98(1)
Chapter 9 Symplectic persistence modules
99(16)
9.1 Liouville manifolds
99(2)
9.2 Symplectic persistence modules
101(2)
9.3 Examples of SH*(U)
103(1)
9.4 Symplectic Banach-Mazur distance
104(2)
9.5 Functorial properties
106(3)
9.6 Applications
109(2)
9.7 Computations
111(4)
Bibliography 115(6)
Notation Index 121(2)
Subject Index 123(4)
Name Index 127
Leonid Polterovich, Tel Aviv University, Israel.

Daniel Rosen, Ruhr-Universitat Bochum, Germany.

Karina Samvelyan, Tel Aviv University, Israel.

Jun Zhang, Universite de Montreal, Canada.
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