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 planning. 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 communities, 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 geometry. 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 of arcs, 205, 206 dl, 172 greedy system of loops, 201 dual, 110 group of q-chains, 56 dual complex, 166 dualizing complex, 96 Hénon map, 67 higher direct image, 94 elementary cell, 60 homeomorphism, 6 elementary chain, 56 Homeomorphism Problem, 14 elementary contraction, 29 homologous, 16, 82 elementary cube, 52 homology, 15, 82 elementary interval, 52 homology class, 82 empty complex, 151 homology group, 16, 57 essential, 177 homology localization, 213, 216 étale sheaf, 90 homology model, 63 étale space, 90 homotopic, 156 Euclidean space, 6 homotopy, 6 Euler calculus, 75 homotopy basis, 199 Euler characteristic, 15, 77 homotopy Conley index, 48 Euler integral, 79, 97 homotopy equivalence, 6 Euler-Poincaréformula,17 homotopy type, 6 Euler-Poincaré index, 115, 126 horseshoe map, 50 evasive, 180, 182 exact, 86 image, 47 execute, 152 incomplete, 15 executed, 150, 159 indecomposable, 26, 27 exit set, 47 index map, 50 extrinsic, 3 index pair, 49 index quadruple, 49 face, 6 induced topology, 6 face operator, 31 injective resolutions, 92 face poset, 174 interval extension, 46 filtration, 20 intrinsic, 3 finite type, 27 intrinsic volumes, 142 flabby, 94 invariant part, 42, 48, 54 flag complex, 10 inverse image, 91 forward-chaining, 148, 186, 188 isolated invariant set, 48 Fourier transform, 116 isolating block, 47 free, 17 isolating neighborhood, 48, 54 free coface, 58 isospectral set, 116 free face, 57 free pair, 29 join, 152 Frongilo, R., 67 join semi-lattices, 185 full, 52 full cubical set, 52 k-cochain, 92 full shift, 43 Kot-Schaffer growth-dispersal model, 68 fully controllable, 162 functor, 186, 187 landmarks, 11 Leray functor, 49 Gaussian elimination, 18 linear programming, 216 general position, 9 link, 111, 172 generalized kernel, 49 lk, 172 generator, 42, 48 local Euler-Poincaré index, 97 ghost solutions, 45 local Morse data, 126 global sections, 90 long exact sequence, 86 goal-preserving, 185 long exact sequence of the pair, 87 good, 7 loopback actions, 156 good cover, 93 loopback complex, 156 INDEX 231 loopback complexes, 179 polyconvex, 78 loopback graph, 156 poset, 174 Lorenz system, 45 poset of local strategies, 187 Lorenz, Edward Norton, 44 positively invariant part, 54 presheaf, 89 manipulate, 147 prestacks, 89 matroid optimization, 212 projection formula, 111 maximal, 6, 10 pullback, 91, 111 maximum flows, 219 pure, 12 Mayer-Vietoris sequence, 87 pushforward, 91 Mayer-Vietoris sequence for sheaves, 99 microlocal Fourier transform, 115 Quillen Fiber Lemma, 174 minimal nonface, 168 quiver, 26 minimal representation, 52 minimal strategy, 182 Radon transform, 111 minimum cuts, 215 range space, 18 minimum-cost circulation, 219 rank invariant, 24 Minkowski sum, 110 reduction, 18 Möbius complex, 218 relative homology, 83 models, 147 relevant region, 206, 208, 209 Moore, R.E., 45 representable intervals, 46 morphism, 91, 185 representable numbers, 46 Morse, 125 representation, 26, 52, 61 Morse decomposition, 70 representation theory, 26 Morse index, 125 restricted Voronoi region, 9 Morse sets, 70 return time map, 42 motivic measure, 76 ring of coefficients, 17 moves off, 160 Robbin, J., 50 moves off W subject to σ, 173 robot, 147 multidimensional persistence, 23 Salamon, D., 50 multifiltration, 20 scissors equivalence, 79 multivalued map, 47 sd, 155 negatively invariant part, 54 section, 90 neighborhood graph, 10 sections, 89 nerve, 7 selector, 53 non-degenerate, 31 semialgebraic, 78 nondeterministic, 150 semicontinuous, 47 nonevasive, 180, 182 (semi)dynamical system, 42 NP-hard, 203, 212, 213, 214, 217 sensing relation, 101 null space, 18 sensitive dependence on initial conditions, 44 o-minimal structure, 78 sensor supports, 101 one-critical, 20 shaving, 34 open ball, 8 sheaf, 88, 89 open cover, 7 sheafification, 91 open set, 6 shortest homotopic cycles, 209 optimal homology basis, 212 shortest homotopic path, 204 order complex, 174 simplex, 6 orientation, 15 simplices, 6 oriented simplex, 15 simplicial complex, 6, 151 simplicial set, 30 periodic, 42 skeleton, 7 persistence barcode, 22 Smale horesehoe map, 44 persistent homology, 21 Smale, Stephen, 43 plan, 152 Smith diagonalization, 56 planner, 147, 186, 188 Smith normal form, 18 Poincarémap,43 solution, 48, 53 Poincaré section, 43 source, 150, 159 232 INDEX source complex, 165, 187 Wazewski’s ˙ Theorem, 47, 48 sphere, 153, 156, 184 weak feature size, 139 src, 160 weak preimage, 47 stalk, 89 weak witness, 11 start region, 160 witness complex, 11 stationary, 42 witness graph, 11 stochastic, 159, 160 worst-case expected convergence times, 161 strategy, 152, 160 strategy complex, 152, 160 zero-skeleton, 151 strong witness, 11 zigzag, 26 subanalytic, 78 zigzag persistence, 25 subcomplex, 7 subgraph, 170 sublevel sets, 21 subshift of finite type, 67 subspace, 6 support, 57 system of arcs, 203 system of loops, 199 t-translation map, 42 tame, 78 target, 159 target supports, 101 targets, 101, 150 thinning, 34 tidy set, 34 tight octagonal decomposition, 206 topological data analysis, 5 topological invariant, 14 topological space, 6 topological type, 6 topology, 1, 6 torsion, 17 total unimodularity, 217 trace, 102 trajectory (orbit), 42 transition time, 161 tree-cotree decomposition, 201 Trevion, R., 67 trimming, 34 trivial invariant, 15 tunnels, 17 uncertainty, 148 underlying space, 7 underlying undirected graph, 26 unified semidynamical system, 69 upper semicontinuous, 47

Verdier duality, 92 Vietoris-Rips complex, 10 void complex, 151 voids, 17 Voronoi diagram, 9 Voronoi region, 9 VR complex, 10

Warmus, M., 45 Titles in This Series

70 Afra Zomorodian, Editor, Advances in Applied and Computational Topology, 2012 69 Karl Sigmund, Editor, Evolutionary Game Dynamics, 2011 68 Samuel J. Lomonaco, Jr., Editor, Quantum Information Science and Its Contributions to Mathematics, 2010 67.2 Eitan Tadmor, Jian-Guo Liu, and Athanasios Tzavaras, Editors, Hyperbolic Problems: Theory, Numerics and Applications, 2009 67.1 Eitan Tadmor, Jian-Guo Liu, and Athanasios Tzavaras, Editors, Hyperbolic Problems: Theory, Numerics and Applications, 2009 67 Eitan Tadmor, Jian-Guo Liu, and Athanasios E. Tzavaras, Editors, Hyperbolic Problems: Theory, Numerics and Applications, 2009 66 Dorothy Buck and Erica Flapan, Editors, Applications of Knot Theory, 2009 65 L. L. Bonilla, A. Carpio, J. M. Vega, and S. Venakides, Editors, Recent Advances in Nonlinear Partial Differential Equations and Applications, 2007 64 Reinhard C. Laubenbacher, Editor, Modeling and Simulation of Biological Networks, 2007 63 Gestur Olafsson´ and Eric Todd Quinto, Editors, The Radon Transform, Inverse Problems, and Tomography, 2006 62 Paul Garrett and Daniel Lieman, Editors, Public-Key Cryptography, 2005 61 Serkan Ho¸sten, Jon Lee, and Rekha R. Thomas, Editors, Trends in Optimization, 2004 60 Susan G. Williams, Editor, Symbolic Dynamics and its Applications, 2004 59 James Sneyd, Editor, An Introduction to Mathematical Modeling in Physiology, Cell Biology, and Immunology, 2002 58 Samuel J. Lomonaco, Jr., Editor, Quantum Computation, 2002 57 David C. Heath and Glen Swindle, Editors, Introduction to Mathematical Finance, 1999 56 Jane Cronin and Robert E. O’Malley, Jr., Editors, Analyzing Multiscale Phenomena Using Singular Perturbation Methods, 1999 55 Frederick Hoffman, Editor, Mathematical Aspects of Artificial Intelligence, 1998 54 Renato Spigler and Stephanos Venakides, Editors, Recent Advances in Partial Differential Equations, Venice 1996, 1998 53 David A. Cox and Bernd Sturmfels, Editors, Applications of Computational Algebraic Geometry, 1998 52 Vidyadhar S Mandrekar and Pesi R. Masani, Editors, Proceedings of the Norbert Wiener Centenary Congress, 1994, 1997 51 Louis H. Kauffman, Editor, The Interface of Knots and Physics, 1996 50 Robert Calderbank, Editor, Different Aspects of Coding Theory, 1995 49 Robert L. Devaney, Editor, Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets, 1994 48 Walter Gautschi, Editor, Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics, 1995 47 Ingrid Daubechies, Editor, Different Perspectives on Wavelets, 1993 46 Stefan A. Burr, Editor, The Unreasonable Effectiveness of Number Theory, 1992 45 De Witt L Sumners, Editor, New Scientific Applications of Geometry and Topology, 1992 44 Béla Bollobás, Editor, Probabilistic Combinatorics and Its Applications, 1991 43 Richard K Guy, Editor, Combinatorial games, 1991 42 Carl Pomerance, Editor, Cryptology and Computational Number Theory, 1991 41 Roger W. Brockett, Editor, Robotics, 1990 40 Charles R. Johnson, Editor, Matrix Theory and Applications, 1990 39 Robert L. Devaney and Linda Keen, Editors, Chaos and Fractals: The Mathematics Behind the Computer Graphics, 1989 TITLES IN THIS SERIES

38 Juris Hartmanis, Editor, Computational Complexity Theory, 1989 37 Henry J. Landau, Editor, Moments in Mathematics, 1987 36 Carl de Boor, Editor, Approximation Theory, 1986 35 Harry H. Panjer, Editor, Actuarial Mathematics, 1986 34 Michael M. Anshel and William Gewirtz, Editors, Mathematics of Information Processing, 1986 33 Peter H. Young, Editor, Fair Allocation, 1985 32 Robert W. McKelvey, Editor, Environmental and natural resources mathematics, 1985 31 Bhaskarpillai Gopinath, Editor, Computer Communications, 1985 30 SimonA.Levin,Editor, Population Biology, 1984 29 David P. Dobkin, George I. Davida, Michael A. Harrison, Richard A. Demillo, and Richard J. Lipton, Editors, Applied Cryptology, Cryptographic Protocols, and Computer Security Models, 1983 28 Ramanathan Gnanadesikan, Editor, Statistical Data Analysis, 1983 27 Lawrence A. Shepp, Editor, Computed Tomography, 1983 26 Stefan A. Burr, Editor, The Mathematics of Networks, 1982 25 Saul I. Gass, Editor, Operations Research Mathematics and Models, 1981 24 William F. Lucas, Editor, Game Theory and Its Applications, 1981 23 Robert V. Hogg, Editor, Modern Statistics: Methods and Applications, 1980 22 Gene H. Golub and Joseph Oliger, Editors, Numerical Analysis, 1978 21 PeterD.Lax,Editor, Mathematical Aspects of Production and Distribution of Energy, 1977 20 Joseph P. Lasalle, Editor, The Influence of Computing on Mathematical Research and Education, 1974 19 J. T. Schwartz, Editor, Mathematical Aspects of Computer Science, 1967 18 H. Grad, Editor, Magneto-fluid and plasma dynamics, 1967 17 R. Finn, Editor, Applications of nonlinear partial differential equations in mathematical physics, 1965 16 R. Bellman, Editor, Stochastic Processes in Mathematical Physics and Engineering, 1964 15 Nicholas C. Metropolis, Abraham H. Taub, C B Tompkins, and J. Todd, Editors, Experimental arithmetic, high speed computing and mathematics, 1963 14 R. E. Bellman, Editor, Mathematical Problems in the Biological Sciences, 1962 13 G. Birkhoff, R. Bellman, and C. C. Lin, Editors, Hydrodynamic instability, 1962 12 R. Jakobson, Editor, Structure of Language and Its Mathematical Aspects, 1961 11 George D. Birkhoff and E. P. Wigner, Editors, Nuclear reactor theory, 1961 10 R. Bellman and M. Hall, Jr, Editors, Combinatorial analysis, 1960 9 G. Birkhoff and R. E. Langer, Editors, Orbit Theory, 1959 8 L. M. Graves, Editor, Calculus of variations and its applications, 1958 7 L. A. MacColl, Editor, Applied Probability, 1957 6 J. H. Curtiss, Editor, Numerical analysis, 1956 5 Albert E. Heins, Editor, Wave motion and vibration theory, 1954 4 M. H. Martin, Editor, Fluid dynamics, 1953 3 Eric Reissner, Abraham H. Taub, and R. V. Churchill, Editors, Elasticity, 1950 2 Abraham H. Taub, Eric Reissner, and R. V. Churchill, Editors, Electromagnetic theory, 1950 1 Eric Reissner and William Pryer, Editors, Nonlinear problems in mechanics of continua, 1949