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Phd Thesis, Stanford University DISSERTATION TOPOLOGICAL, GEOMETRIC, AND COMBINATORIAL ASPECTS OF METRIC THICKENINGS Submitted by Johnathan E. Bush Department of Mathematics In partial fulfillment of the requirements For the Degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Summer 2021 Doctoral Committee: Advisor: Henry Adams Amit Patel Chris Peterson Gloria Luong Copyright by Johnathan E. Bush 2021 All Rights Reserved ABSTRACT TOPOLOGICAL, GEOMETRIC, AND COMBINATORIAL ASPECTS OF METRIC THICKENINGS The geometric realization of a simplicial complex equipped with the 1-Wasserstein metric of optimal transport is called a simplicial metric thickening. We describe relationships between these metric thickenings and topics in applied topology, convex geometry, and combinatorial topology. We give a geometric proof of the homotopy types of certain metric thickenings of the circle by constructing deformation retractions to the boundaries of orbitopes. We use combina- torial arguments to establish a sharp lower bound on the diameter of Carathéodory subsets of the centrally-symmetric version of the trigonometric moment curve. Topological information about metric thickenings allows us to give new generalizations of the Borsuk–Ulam theorem and a selection of its corollaries. Finally, we prove a centrally-symmetric analog of a result of Gilbert and Smyth about gaps between zeros of homogeneous trigonometric polynomials. ii ACKNOWLEDGEMENTS Foremost, I want to thank Henry Adams for his guidance and support as my advisor. Henry taught me how to be a mathematician in theory and in practice, and I was exceedingly fortu- nate to receive my mentorship in research and professionalism through his consistent, careful, and honest feedback. I could always count on him to make time for me and to guide me to interesting problems. I would like to thank my undergraduate advisor, George McRae, for sharing with me the joy of problem solving. I look back fondly on many hours spent standing in his office and sketching with colored chalk the commutative diagrams he would dictate to me from his cluttered desk. I want to thank Florian Frick for introducing me to topological methods in combinatorics and geometry. The results in Chapter 5 and Chapter 6 are joint with him, and many of the ideas underpinning the connections between metric thickenings and orbitopes described in Chapter 3 are due to him. These ideas have had a lasting impact on my approach to applied mathematics. I would also like to thank Harrison Chapman, Michael Crabb, Facundo Mémoli, Isabella Novik, Amit Patel, Chris Peterson, and Hailun Zheng for helpful conversations. I am especially grateful to my fiancée, Jordan, for her constant support and for helping me navigate my challenges and accomplishments with patience, perspective, and grace. I am grate- ful to my office mates Alex, Dean, Dustin, Elliot, Joshua, Levi, and Scott for their camaraderie and friendship. Finally, I am grateful to my parents, Harley and Dorothy, for their reassurance and faith in my ability to succeed. iii DEDICATION to my family iv TABLE OF CONTENTS ABSTRACT................................................ ii ACKNOWLEDGEMENTS . iii DEDICATION . iv LIST OF FIGURES . vii Chapter 1 Introduction . 1 Chapter 2 Background and related work . 3 2.1 Applied topology . 3 2.1.1 Simplicial metric thickenings and optimal transport . 10 2.1.2 Recovery of a metric space from a simplicial complex . 12 2.1.3 The persistent homology of manifolds . 17 2.2 Convex geometry and orbitopes . 21 2.2.1 Conventions regarding spheres and circular arcs . 23 2.2.2 Trigonometric polynomials . 23 2.2.3 The trigonometric moment curve . 24 2.2.4 The centrally symmetric trigonometric moment curve . 24 2.2.5 Orbitopes . 25 2.2.6 Carathéodory orbitopes . 26 2.2.7 Barvinok–Novik orbitopes . 29 2.2.8 Vandermonde matrices and related matrices . 32 2.3 The Borsuk–Ulam theorem . 34 2.3.1 Corollaries of the Borsuk–Ulam theorem . 35 Chapter 3 Metric thickenings of the circle . 40 3.1 Carathéodory and Barvinok–Novik metric thickenings . 40 3.2 Cechˇ and Vietoris–Rips metric thickenings of the circle . 42 3.2.1 Outline of the proof technique . 43 3.2.2 Map from the metric thickening to the boundary of an orbitope . 45 3.2.3 Map from the boundary of an orbitope to the metric thickening . 46 3.2.4 Show ¶ is a homotopy equivalence . 47 Chapter 4 Carathéodory subsets of moment curves and the faces of Carathéodory and Barvinok–Novik orbitopes . 55 4.1 Carathéodory subsets of the trigonometric moment curve . 55 4.2 Carathéodory subsets of the centrally symmetric trigonometric moment curve......................................... 56 4.2.1 The proof of Theorem 4.2.1 . 56 4.3 The Carathéodory coorbitope cone . 63 4.4 The Barvinok–Novik coorbitope cone . 68 Chapter 5 Generalizations of the Borsuk–Ulam theorem . 72 v 5.1 Maps S1 Rk with k 1 ............................. 72 ! È 5.2 Maps Sn Rk with k n.............................. 75 ! È 5.3 Corollaries . 80 5.3.1 Generalization of the Stone–Tukey theorem . 81 5.3.2 Generalization of the Lyusternik–Shnirel’man–Borsuk theorem . 83 5.3.3 Traversals of bounded diameter . 84 Chapter 6 Zeros of trigonometric polynomials . 86 Chapter 7 Conclusion . 91 Bibliography . 93 Appendix A Proof of Proposition 4.3.1 . 104 vi LIST OF FIGURES 2.1.1 (Top left) A discrete subset X R2. (Top right) Balls of radius 1 r centered at the ½ 2 points of X . (Bottom left) The ambient Cechˇ complex at scale r . (Bottom right) The Vietoris–Rips complex at scale r . ............................... 5 2.1.2 (Top) A discrete metric space and its Vietoris–Rips complexes at five different choices of scale. (Bottom) The 0-dimensional and 1-dimensional persistent homology inter- vals. The horizontal axis is the scale parameter r in the construction of the Vietoris– Rips complex. 9 2.2.1 Examples of non-negative trigonometric polynomials defining faces of Carathéodory orbitopes. A formula for the coefficient vectors of all such polynomials in given in Section 4.3. 27 2.2.2 (Left) The image of the map S1 R3 defined by t (cos(t),sin(t),cos(2t)). (Right) ! 7! The convex hull of this set. 28 2.2.3 (Left) The image of the map S1 R3 defined by t (cos(t),sin(t),cos(3t)). (Right) ! 7! The convex hull of this set. 30 2.3.1 Translated, this footnote reads “This theorem was posed as a conjecture by St. Ulam.” 34 2.3.2 A 3-coloring of the Petersen graph K (5,2). 38 SM p 3.2.1 The composition of maps VRm(S1;r ) 2k R2k \{~0} @B , drawn in the case k 1. 45 ¡¡¡! ¡! 2k Æ 3.2.2 An example of points {γ1,γ2} and {v1,...,v3}, as defined in the proof of Proposi- tion 3.2.12, that are used to construct a vector satisfying the hypotheses of Lemma 3.2.9. 51 4.3.1 A set of non-negative trigonometric polynomials f~s, as defined in Example 4.3.7, each of which defines a 1-dimensional face on the boundary of the Carathéodory orbitope C4. The non-zero root of each polynomial has been chosen at random. 67 4.4.1 Non-negative trigonometric polynomials g~s, as defined in Equation 4.2, each of which defines a face of the Barvinok–Novik orbitope B4. For clarity, each polynomial has been multiplied by a non-zero constant to achieve the same maximum value. 71 5.1.1 A subset of S1 of small diameter whose image is a Carathéodory subset under an odd map................................................. 74 5.3.1 (Left) A bundle of three logs. Dashed blue lines indicate horizontal cuts. (Right) A vertical cut through the center of one slice of the log bundle. In this case, the saw 2¼ blade is on a fixed pivot that can not swivel by an angle of more than 3 ......... 81 vii Chapter 1 Introduction Given a metric space X , one may embed X into P (X ), the space of Radon probability mea- sures on X , by mapping a point to the Dirac delta measure centered at that point. This em- bedding is an isometry onto its image. One may thicken this image inside P (X ) by first tak- ing convex combinations of points and then equipping the resulting set with the restriction of the 1-Wasserstein metric of optimal transport. In the case that these convex combinations are constrained to support sets of a simplicial complex on X such that all simplices have diameter bounded above by r 0, the resulting space is a metric r -thickening of X in the sense of Gromov ¸ and is called a simplicial metric thickening. This thesis is an attempt to expose and elucidate mathematical threads connecting simpli- cial metric thickenings and topics in applied topology and convex geometry, and, to a lesser extent, algebraic topology and topological combinatorics. Along these lines, we explain con- nections between Cechˇ and Vietoris–Rips metric thickenings and convex bodies in Euclidean space, odd maps with large Carathéodory subsets, generalizations of the Borsuk–Ulam theo- rem, and quantitative results about the distribution of roots of trigonometric polynomials. We are hopeful the results in this thesis will demonstrate the utility of this multi-faceted approach to the study of metric thickenings. In Chapter 2, we summarize background material and related work in applied and pure topology and convex geometry. We review Cechˇ and Vietoris–Rips simplical complexes, sim- plicial metric thickenings defined by the optimal transport metric, reconstruction results for manifolds, and results about the persistent homology of manifolds. We fix notation and defini- tions from convex geometry and describe families of convex bodies, called orbitopes, that will play a key role in the sequel.
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