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Persistence, Metric Invariants, and Simplification Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Osman Berat Okutan, M.S. Graduate Program in Mathematics The Ohio State University 2019 Dissertation Committee: Facundo M´emoli,Advisor Matthew Kahle Jean-Franc¸oisLafont c Copyright by Osman Berat Okutan 2019 Abstract Given a metric space, a natural question to ask is how to obtain a simpler and faithful approximation of it. In such a situation two main concerns arise: How to construct the approximating space and how to measure and control the faithfulness of the approximation. In this dissertation, we consider the following simplification problems: Finite approx- imations of compact metric spaces, lower cardinality approximations of filtered simplicial complexes, tree metric approximations of metric spaces and finite metric graph approxima- tions of compact geodesic spaces. In each case, we give a simplification construction, and measure the faithfulness of the process by using the metric invariants of the original space, including the Vietoris-Rips persistence barcodes. ii For Esra and Elif Beste iii Acknowledgments First and foremost I'd like to thank my advisor, Facundo M´emoli,for our discussions and for his continual support, care and understanding since I started working with him. I'd like to thank Mike Davis for his support especially on Summer 2016. I'd like to thank Dan Burghelea for all the courses and feedback I took from him. Finally, I would like to thank my wife Esra and my daughter Elif Beste for their patience and support. This research was supported by NSF grants AF 1526513, DMS 1723003, CCF 1740761, IIS-1422400 and CCF-1526513 . iv Vita May 2010 . .B.S. in Mathematics Bilkent University May 2012 . .M.S. in Mathematics Bilkent University September 2012 - present . Graduate Teaching Associate Department of Mathematics The Ohio State University. Publications Research Publications O. Okutan, E. Yalcin \Free actions on products of spheres at high dimensions". Algebraic & Geometric Topology 13, pp: 2087-2099, 2013. Fields of Study Major Field: Mathematics Specialization: Topological Data Analysis, Metric Geometry v Table of Contents Page Abstract . ii Dedication . iii Acknowledgments . iv Vita.............................................v List of Figures . ix 1. Introduction . .1 1.1 Content . .3 2. Background . .9 2.1 Metric Geometry . .9 2.1.1 Gromov-Hausdorff Distance . 10 2.1.2 Hyperbolicity . 15 2.1.3 Injective/Hyperconvex Spaces . 17 2.2 Persistence . 19 2.2.1 Persistence Sequence . 22 2.3 Interleaving Distance . 27 2.4 Metric Graphs . 28 2.5 Reeb Graphs . 35 2.5.1 Stability of Reeb Metric Graphs . 39 2.5.2 Smoothings . 40 2.6 Differential Topology . 42 vi 3. A Geometric Characterization of Vietoris-Rips Filtration . 46 3.1 Introduction . 46 3.2 Persistence via Metric Pairs . 48 3.3 Isomorphism . 52 3.3.1 Stability of Metric Homotopy Pairings . 54 3.4 Application to the Vietoris-Rips Filtration . 56 3.4.1 Products and Wedge Sums . 56 3.5 Applications to the Filling Radius . 60 3.5.1 Bounding Barcode Length via Spread . 60 3.5.2 Bounding the Filling Radius . 62 3.5.3 Stability . 64 4. Finite Approximations of Compact Metric Spaces . 67 4.1 Introduction . 67 4.2 Discrete Length Structures . 68 4.3 Discrete Length structures on Metric Spaces . 69 4.4 Proof of Theorem 4.1.1 . 73 5. Metric Graph Approximations of Geodesic Spaces . 75 5.1 Introduction . 75 5.2 Graph Approximations . 79 5.3 Tree Approximations . 82 6. The Distortion of the Reeb Quotient Map on Riemannian Manifolds . 92 6.1 Introduction . 92 6.2 Distortion of the Reeb Quotient Map . 95 6.3 Thickness . 97 6.3.1 A Calculation: Thickened Filtered Graphs . 99 6.4 The Diameter of a Fiber of the Reeb Quotient Map . 103 6.5 The Bound of Theorem 6.1.1 for Thickened Graphs . 105 7. Reeb Posets and Metric Tree Approximations . 109 7.1 Introduction . 109 7.2 Posets . 111 7.3 Reeb Constructions . 115 7.3.1 Poset Paths and Length Structures . 115 vii 7.3.2 Reeb Posets . 117 7.3.3 Reeb Tree Posets . 119 7.4 Hyperbolicity for Reeb Posets . 122 7.5 Approximation . 124 7.6 An Application to Metric Graphs and Finite Metric Spaces . 126 7.7 Example where Φ ∼ Υ............................. 129 8. Quantitative Simplification of Filtered Simplicial Complexes . 132 8.1 Introduction . 132 8.2 Gromov-Hausdorff and Interleaving Type Distances between Filtered Sim- plicial Complexes . 139 8.2.1 Gromov-Hausdorff Distance between Filtered Simplicial complexes 139 F 8.2.2 The Interleaving Type Distance dI between Filtered Simplicial Com- plexes . 143 F 8.2.3 Remarks About the Definition of dI .................. 149 8.2.4 Stability and the Proof of Theorem 8.1.1 . 150 8.3 The Vertex Quasi-distance and Simplification . 152 8.3.1 Computing δX (v; w): The Procedure ComputeCodensityMatrix() . 155 8.3.2 Specializing δX (v; w) According to Homology Degree . 158 8.4 An Application to the Vietoris-Rips Filtration of Finite Metric Spaces and Graphs . 160 8.4.1 Finite Metric Spaces . 160 8.4.2 Application to Metric Graphs . 163 F 8.5 Classification of Filtered Simplicial Complexes via dI ............ 168 F 8.6 An Example where dI dGH ......................... 171 8.7 Chain Construction . 173 9. Future work . 176 Bibliography . 178 viii List of Figures Figure Page 1.1 The space on the right is a simplification of the space on the left. .2 1.2 B(X; r) is the r-neighborhood of X in E. Notice how the small loop in X is filled. .3 1.3 A four point approximation of a metric space with three components. .4 2.1 A metric graph . 29 2.2 Reeb graph Xh of the height function h on X.................. 35 3.1 A big sphere X with a small handle. In this case, as r > 0 increases, 2 Br(X; κ(X)) changes homotopy type from that of X to that of S as soon as r > r0 for some r0 < FillRad(X): ....................... 63 2 ab a a 6.1 TA = a2+b2 depends only on b and converges to 0 as b ! 0. 98 6.2 A 2-dimensional thickened filtered graph. 99 6.3 A 2-dimensional thickened 3-fork. 100 6.4 An inverse 2-dimensional thickened fork. 102 6.5 A vertical fork. 106 7.1 A finite metric space embedded in a metric graph . 110 7.2 Let R ≥ r > 0 and consider the metric graph from the figure. Let Zn be the fi- nite subset fp; x0; : : : ; xn; y1; : : : ; yng: We show that Φ(Zn) ∼ 2 log(4n) hyp(Zn) and Υp(Zn) = 2 log(4n + 4) hyp(Zn). ...................... 130 ix 8.1 These two finite spaces have the same Vietoris-Rips PH≥1, see Example 8.4.3. 133 ∗ ∗ 8.2 X := ∆3: ..................................... 148 8.3 A simple metric graph. 166 F 8.4 These two filtered simplicial complexes are at 0 dI -distance while they are at 1 Gromov-Hausdorff distance at least 2 ...................... 172 x Chapter 1: Introduction In Data Analysis, a data set can generally be endowed with a metric structure. This enables the analysis of the data set not just from a statistical point of view, but also from a geometric point of view. Topological Data Analysis tries to combine and take advantage of both the quantitative (but albeit often noisy) nature of Data and the qualitative nature of Topology [19]. Geometric intuition tells us that metric spaces have topological features and these features have quantitative properties like size. To be able to recognize these quantitative properties, one needs to go beyond the topological structure induced by the metric, since a given topo- logical space can be metrized in many different ways. Furthermore, as an example, if we consider a finite metric subspace of a unit circle, it is supposed to have an inherited circularity property as a metric space, but the underlying topological space is just discrete. Similarly, if we only look at the underlying topology, finite metric spaces have discrete topology, which can not explain any expected topological feature of the metric space. The main insight of persistence ([69, 38, 19]) for metric spaces is the following: A metric space does not simply induce a topological space, but a family of topological spaces indexed by non-negative real numbers. This family generally arises as a filtration. Then one can observe how topological features change as we change the index. 1 Figure 1.1: The space on the right is a simplification of the space on the left. Given a metric space X, the most common way of obtaining a family of topological spaces r r is via the Vietoris-Rips filtration (VR (X))r≥0 , where VR (X) is the simplicial complex with the vertex set X and simplices given by finite subsets of X with diameter less than or equal to r. There is also a more geometric method of obtaining a filtration, which is equivalent to the Vietoris-Rips filtration up to homotopy. Given a metric space X, there are several natural metric spaces into which X can.
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