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Advances in Applied and Computational Topology, Volume 70 http://dx.doi.org/10.1090/psapm/070 AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures published as a subseries of Proceedings of Symposia in Applied Mathematics This page intentionally left blank Proceedings of Symposia in APPLIED MATHEMATICS Volume 70 Advances in Applied and Computational Topology AMS Short Course Computational Topology January 4–5, 2011 New Orleans, Louisiana Afra Zomorodian Editor American Mathematical Society Providence, Rhode Island Editorial Board Mary Pugh Lenya Ryzhik Eitan Tadmor (Chair) LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE COMPUTATIONAL TOPOLOGY HELD IN NEW ORLEANS, LOUISIANA JANUARY 4–5, 2011 The AMS Short Course Series is sponsored by the Society’s Program Committee for National Meetings. The series is under the direction of the Short Course Subcommittee of the Program Committee for National Meetings. 2010 Mathematics Subject Classification. Primary 55N35, 55U05, 55–04, 37B30, 37M99, 37D45, 55N30, 53C65, 68T37, 68T40, 68W05, 68Q25. Library of Congress Cataloging-in-Publication Data American Mathematical Society. Short Course on Computational Topology (2011 : New Orleans, La.) Advances in applied and computational topology : American Mathematical Society Short Course on Computational Topology, January 4–5, 2011, New Orleans, Louisiana / Afra Zomoro- dian, editor. p. cm. — (Proceedings of symposia in applied mathematics ; v. 70) Includes bibliographical references and index. ISBN 978-0-8218-5327-6 (alk. paper) 1. Algebra, Homological—Congresses. 2. Homology theory—Congresses. 3. Ergodic theory— Congresses. I. Zomorodian, Afra J., 1974– II. Title. QA169.A475 2011 514—dc23 2012008031 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. 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 http://www.ams.org/ 10987654321 171615141312 To Dr. Benjamin Mann This page intentionally left blank Contents Preface ix Topological Data Analysis Afra Zomorodian 1 Topological Dynamics: Rigorous Numerics via Cubical Homology Marian Mrozek 41 Euler Calculus with Applications to Signals and Sensing Justin Curry, Robert Ghrist, and Michael Robinson 75 On the Topology of Discrete Planning with Uncertainty Michael Erdmann 147 Combinatorial Optimization of Cycles and Bases Jeff Erickson 195 Index 229 vii This page intentionally left blank Preface This volume is the proceedings of the AMS Short Course on Computational Topology, organized for the Joint Mathematics Meetings in New Orleans on Janu- ary 4 – 5, 2011. Computational topology emerged in response to topological imped- iments within geometric problems, such as extraneous holes and tunnels in surfaces reconstructed by the computer graphics and computational geometry communities. Topological problems arise naturally, however, in many areas of science. In robotics, we need to capture the connectivity of the configuration space of a robot for plan- ning. In sensor networks, we wish to deduce global information from local sensing. In dynamical systems, we want to understand qualitative properties of a system via computation. In data analysis, we look for robust features of an underlying space given a finite set of noisy samples. Like most emerging areas, computational topology is claimed by several com- munities, and naturally, each community defines the area in its own image. With this course, I wanted to broaden the definition of this field to include any area that resolves a topological question using computational techniques. To this end, I invited speakers from a broad spectrum of specializations, including algebraic topology, dynamical systems, applied topology, robotics, and computational ge- ometry. I also wanted the course to cover the rich development of computational topology from theory, to algorithm design and analysis, implementation of fast software, and applications. The structure of the book mirrors that of the course. We dedicated the first day to topological data analysis. One of the speakers, Gunnar Carlsson, did not contribute a chapter, as he is currently developing a book on the subject. The day culminated with a software session: Henry Adams provided a tutorial on JPlex,a Java software package for topological data analysis; Marian Mrozek demonstrated RedHom,aC++ library for computing cubical homology. The second day of the short course concentrated on applications of topology to sensor networks, robotics, and geometry. Robert Ghrist contributed his chapter in collaboration with two colleagues. The short course concluded with a panel session, during which the speakers and the attendees discussed the state and future of computational topology. Ten years ago, publishing in this area was difficult as the required mathematics was unknown to computer scientists, while the value of the applications was unappreciated by mathematicians. By now, a number of conferences and journals have recognized computational topology as a subarea. As it matures through ad-hoc workshops and programs, computational topology will require dedicated conferences and journals so that researchers have centralized forums for disseminating their research. ix xPREFACE I am grateful to Dan Rockmore for soliciting the short course proposal and for his many helpful comments. I thank the speakers for their excellent presentations as well as developing their contributed chapters during the last year. The book went through a two-stage peer-review process in which the speakers and anony- mous reviewers participated. I thank all reviewers for their on-time and thorough critiques. Finally, I thank Sergei Gelfand and Christine Thivierge from the Ameri- can Mathematical Society for shepherding this project. During his tenure at DARPA, Dr. Benjamin Mann was an energetic champion of researchers in applied and computational topology. We dedicate this volume to him in deep appreciation of his persistent support. Afra Zomorodian February 2012 New York, NY Index F(Σ), 174 cochain complex, 83 G←s, 156 cocycle condition, 112 Sm, 153 cocycles, 83 Sn−2, 156, 157, 162 coefficients, 16 Z2-homology cover, 214 coface, 6 Δ , 152, 160 cofaces, 83 ∗G ΔG, 166 cohomological Conley index, 48 ΔG, 165 cohomologous, 83 n = , 157 cohomology, 83 , 156 cohomology sheaf, 94 n = , 157 cohomology with compact supports, 83 collapse, 32 action, 150, 159 combinatorial enclosure, 53 acyclic, 33 combinatorial index pair, 54 acyclic-valued, 61 combinatorial map, 52 additive, 79 combinatorial Morse decomposition, 70 adjunction, 92 combinatorial Morse sets, 70 admissible order, 70 combinatorial surfaces, 197 adv, 161 commutative diagram, 86 adversity, 161 complete invariant, 15 Alexander-Spanier cohomology, 50 complete strategy, 156, 157 alpha and omega limit sets, 42 components, 17 alpha complex, 9 conformation space, 2 analysis, 2 Conley index, 50 associated digraph, 52 Conley, Charles, 48 backchain, 173, 188 Conley-Morse graph, 70 backchaining, 148 connecting map, 87 barycentric subdivision, 155 constructible, 79 base change, 100 constructible feature size, 139 Bessel transform, 117 constructible sheaf, 96 Betti number, 17 continuation classes, 71 boundary, 16 continuation graph, 71 boundary homomorphism, 16, 33 continuous, 6 boundary operator, 57 contractible, 6, 157 contravariant, 89 CAPD, 64 convergent, 152, 160 CAPD and CAPD::RedHom software convolution, 110 projects, 64 covering set, 162 CAPD::RedHom, 64 crossing sequence, 204, 206 categorification, 81 cubical complex, 12 category, 184 cubical product, 56 category of endomorphisms, 49 cycle, 16, 82 Cechˇ cohomology, 92 cycle-preserving, 185 Cechˇ complex, 8 certainly attainable, 162 data, 2 chain complex, 16 Day, S., 67 chain group, 16, 33 decision tree, 180, 181 chain map, 85 definable, 78 chain selector, 61 deformation retract, 47 circuit, 152, 160 degeneracy operator, 31 classifying space, 184, 186, 188 degenerate, 31 clique, 10 Delaunay complex, 9 clique complex, 10 deletion, 172 closed, 6 derived categories, 94 co-index, 126 design, 149, 168 coboundary, 83 dimension, 6, 17, 52, 56 229 230 INDEX direct image, 91 grading, 82 discrete Morse theory, 65 graph, 148, 150, 160 discrete semidynamical system, 42 greedy system
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