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Applied VOLUME 74 UNIVERSITY LECTURE SERIES Mathematics Topological Persistence in Geometry and Analysis Leonid Polterovich Daniel Rosen Karina Samvelyan Jun Zhang 10.1090/ulect/074 Topological Persistence in Geometry and Analysis UNIVERSITY LECTURE SERIES VOLUME 74 Topological Persistence in Geometry and Analysis Leonid Polterovich Daniel Rosen Karina Samvelyan Jun Zhang EDITORIAL COMMITTEE Robert Guralnick William P. Minicozzi II (Chair) Emily Riehl Tatiana Toro 2010 Mathematics Subject Classification. Primary 55U99, 58Cxx, 53Dxx. For additional information and updates on this book, visit www.ams.org/bookpages/ulect-74 Library of Congress Cataloging-in-Publication Data Names: Polterovich, Leonid, 1963- author. | Rosen, Daniel, 1980- author. | Samvelyan, Karina, 1988- author. | Zhang, Jun, 1988- author. Title: Topological persistence in geometry and analysis / Leonid Polterovich, Daniel Rosen, Karina Samvelyan, Jun Zhang. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Univer- sity lecture series, 1047-3998 ; volume 74 | Includes bibliographical references and index. Identifiers: LCCN 2019059052 | ISBN 9781470454951 (paperback) | ISBN 9781470456795 (ebook) Subjects: LCSH: Algebraic topology. | Homology theory. | Combinatorial topology. | Symplectic geometry. | Mathematical analysis. | AMS: Algebraic topology – Applied homological alge- bra and category theory [See also 18Gxx]. | Global analysis, analysis on manifolds [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx] {For geometric integration theory, see 49Q15} – Calculus on manifolds; nonlinear operators [See also 46Txx]. | Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx} – Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx]. Classification: LCC QA612 .P645 2020 | DDC 514/.2–dc23 LC record available at https://lccn.loc.gov/2019059052 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2020 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 252423222120 Contents Preface vii Part 1. A primer of persistence modules 1 Chapter 1. Definition and first examples 3 1.1. Persistence modules 3 1.2. Morphisms 5 1.3. Interleaving distance 7 1.4. Morse persistence modules and approximation 9 1.5. Rips modules and the Gromov-Hausdorff distance 9 Chapter 2. Barcodes 13 2.1. The Normal Form Theorem 13 2.2. Bottleneck distance and the Isometry Theorem 19 2.3. Corollary: Stability theorems 20 2.4. Persistence modules of locally finite type 20 Chapter 3. Proof of the Isometry Theorem 23 3.1. An outline 23 3.2. Matchings for surjections and injections 24 3.3. Main lemmas and proof of the theorem 29 3.4. Proofs of Lemma 3.3.1 and Lemma 3.3.2 30 Chapter 4. What can we read from a barcode? 33 4.1. Infinite bars and characteristic exponents 33 4.2. Boundary depth and approximation 36 4.3. The multiplicity function 39 4.4. Representations on persistence modules 41 Part 2. Applications to metric geometry and function theory 47 Chapter 5. Applications of Rips complexes 49 5.1. δ - hyperbolic spaces 49 5.2. Cechˇ complex, Rips complex and topological data analysis 54 5.3. Manifold learning 57 Chapter 6. Topological function theory 61 6.1. Prologue 61 6.2. Invariants of upper triangular matrices 63 6.3. Simplex counting method 66 v vi CONTENTS 6.4. The length of the barcode 68 6.5. Approximation by trigonometric polynomials 71 Part 3. Persistent homology in symplectic geometry 75 Chapter 7. A concise introduction to symplectic geometry 77 7.1. Hamiltonian dynamics 77 7.2. Symplectic structures on manifolds 78 7.3. The group of Hamiltonian diffeomorphisms 79 7.4. Hofer’s bi-invariant geometry 80 7.5. A short tour in coarse geometry 82 7.6. Zoo of symplectic embeddings 83 Chapter 8. Hamiltonian persistence modules 87 8.1. Conley-Zehnder index 87 8.2. Filtered Hamiltonian Floer theory 89 8.3. Constraints on full powers 94 8.4. Non-contractible closed orbits 96 8.5. Barcodes for Hamiltonian homeomorphisms 98 Chapter 9. Symplectic persistence modules 99 9.1. Liouville manifolds 99 9.2. Symplectic persistence modules 101 9.3. Examples of SH∗(U) 103 9.4. Symplectic Banach-Mazur distance 104 9.5. Functorial properties 106 9.6. Applications 109 9.7. Computations 111 Bibliography 115 Notation Index 121 Subject Index 123 Name Index 127 Preface The theory of persistence modules is an emerging field of algebraic topology, which originated in topological data analysis and which lies at the crossroads of several disciplines, including metric geometry and the calculus of variations. Per- sistence modules were introduced by G. Carlsson and A. Zomorodian [110] in 2005 as an abstract algebraic language for dealing with persistent homology, a version of homology theory invented by H. Edelsbrunner, D. Letscher, and A. Zomorodian [32] at the beginning of the millennium, aimed, in particular, at extracting robust information from noisy topological patterns. We refer to the articles by H. Edels- brunner and J. Harer [31], R. Ghrist [39], G. Carlsson [17], S. Weinberger [104], and U. Bauer and M. Lesnick [8] and the monographs by H. Edelsbrunner [30], S. Oudot [76], and F. Chazal, V. de Silva, M. Glisse, and S. Oudot [20] for various aspects of this rapidly developing subject. In the past few years, the theory of persistence modules expanded its “sphere of influence” within pure mathematics, exhibiting fruitful interactions with function theory and symplectic geometry. The purpose of these notes is to provide an introduction to this field and to give an account on some of the recent advances, emphasizing applications to geometry and analysis. We tried to minimize algebraic prerequisites so that the material should be accessible to readers with a basic background in algebraic and differential topology. More advanced preliminaries in geometry and function theory will be reviewed. Topological data analysis deals with data clouds modeled by finite metric spaces. Its main motto is geometry + scale = topology. In the case when a finite metric space appears as a discretization of a Riemannian manifold M, the above relation enables one to infer the topology of M, provided one knows the mesh. In general, given a scale t>0, one can associate to any abstract metric space (X, d) a topological space Rt, called the Vietoris-Rips complex.By definition, Rt is a subcomplex of the full simplex Σ formed by the points of X,where σ ⊂ X is a simplex of Rt whenever the diameter of σ is <t. For instance, the Rips complex for the vertices of the unit square in the plane is presented in Figure 1 a). Thus we get a filtered topological space, a.k.a., a collection of topological spaces Rt, t ∈ R, with Rs ⊂ Rt for s<t. Let us mention that Rt is empty for t ≤ 0and Rt =Σfort>diam(X, d). Some authors call this structure a topological signature of the data cloud (X, d). Rips complexes, which originated in geometric group theory [11], also play an important role in detecting low-dimensional topological patterns in big data, which nowadays is an active area of applied mathematics (see e.g. [73]). The calculus of variations studies critical points and critical values of func- tionals, the simplest case being smooth functions on manifolds. The sublevel sets vii viii PREFACE Figure 1. The Rips complex of a square and the corresponding barcode. = S2 H2 max max = D2 b b 1 = S × R H1 a a = D2 min min ∅ H0 a) b) Figure 2. The height function on the (topological) sphere and the corresponding barcode. Rt := {f<t} of a function f on a closed manifold M induce a structure of a filtered topological space. According to Morse theory, the topology of Rt, t ∈ R, changes exactly when the parameter t hits a critical value of f.NotethatRt = ∅ when t ≤ min f and Rt = M when t>max f. See Figure 2 a) illustrating sublevel sets of a function on the two-dimensional sphere with two local maxima and one local minima. We are going to study filtered topological spaces by using algebraic tools. Fix afieldF and look at the homology Vt := H(Rt, F) of the spaces Rt as above with coefficients in F. The family of vector spaces Vt, t ∈ R together with the morphisms Vs → Vt, s<tinduced by the inclusions, form an algebraic object called a persistence module, which plays a central role in the present notes. Let us discuss the contents of the book in more detail. Part 1 lays foundations of the theory of persistence modules and introduces basic examples. It turns out that persistence modules (which are defined in Chapter 1) are classified by simple combinatorial objects, the barcodes, which are defined as collections of intervals and rays in R with multiplicities. While the real meaning of barcodes will be clarified PREFACE ix later on, some intuition can be gained by looking at Figures 1 b) and 2 b).
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