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Intro; Preface; A Brief Introduction to Computational Topology; Working Groups; Contributed Papers; Final Remarks; References; Contents; The Rank Invariant Stability via Interleavings; 1 Introduction; 2 Background Definitions; 3 A Lower Bound for the Interleaving Distance; 4 Internal Stability of the Rank Invariant; 5 Conclusions; References; Persistent Homology over Directed Acyclic Graphs; 1 Introduction; 2 Definition; 2.1 Graph Filtrations; 2.2 Persistence Module; 2.3 Persistent Homology; 2.4 Barcodes and Carrier Subgraphs; 2.5 An Example; 3 Properties of Persistence Module
3.1 Fundamental Properties3.2 Relationship with Other Models of Persistence; 4 Single-Source Single-Sink Subgraphs; 5 Indecomposable Submodules and Persistence over General Subgraphs; 6 Applications; 6.1 Estimating Persistence Using Multiple Subsamples; 6.2 Shape Comparison; 7 Future Work; References; A Complete Characterization of the One-Dimensional Intrinsic Čech Persistence Diagrams for Metric Graphs; 1 Introduction; 2 Background; 2.1 Homology; 2.2 Persistent Homology and Metric Graphs; 3 From Graphs to Intrinsic Čech Complexes; 3.1 Overview and Relevant Notations
3.2 Relating Graphs to Intrinsic Čech Complexes4 Proof of Main Theorem; 5 Future Work; References; Comparing Directed and Weighted Road Maps; 1 Introduction; 2 Related Work; 2.1 Weighted and Directed Graph Comparison; 2.2 Persistence-Based Comparisons; 3 Filtrations; 3.1 Swatch Filtration; 3.2 Average Minimum Distance; 3.3 Killing Cycles; 4 Comparing Data on Graphs; 4.1 Paired Analysis; 4.2 Birth-Birth Diagrams; 4.3 Examples; 5 Discussion; References; Sweeping Costs of Planar Domains; 1 Introduction; 2 Related Work; 3 Preliminaries and Notation; 3.1 Jordan Domains and Geodesics
3.2 Fréchet and Geodesic Fréchet Distances3.3 Sensor Curves; 4 Properties of Sensor Sweeps; 5 A Lower Bound on the Sweeping Cost; 6 A Lemma of No Progress; 7 Sweeping Cost of a Jordan Domain; 8 Sweeping Cost of a Convex Domain; 9 Extremal Shapes; 10 Conclusion; Appendix: Additional Lemmas and Proofs; References; Scaffoldings and Spines: Organizing High-Dimensional Data Using Cover Trees, Local Principal Component Analysis, and Persistent Homology; 1 Introduction; 1.1 Outline of Method and Paper; 1.2 Related Work; 2 Background; 2.1 Graph Theory; 2.2 Cover Tree; 2.3 Stratified Spaces
2.4 Local Principal Component Analysis2.5 Persistent Homology; 3 The Adaptive Cover Tree and the Scaffolding; 3.1 Eigenmetric Threshold; 3.2 Building the Scaffolding; 4 Local Dimension Estimation and Strata Determination; 4.1 Local Dimension Estimation; 4.2 Topology of the Strata; 5 The Spine; 6 Experiments; 6.1 Synthetic Examples; 6.2 Two Experiments on Music Structure Visualization; 7 Discussion; References; Density of Local Maxima of the Distance Function to a Set of Points in the Plane; 1 Introduction; 2 Main Result; 3 Construction; 4 Discussion and Open Problems; References
3.1 Fundamental Properties3.2 Relationship with Other Models of Persistence; 4 Single-Source Single-Sink Subgraphs; 5 Indecomposable Submodules and Persistence over General Subgraphs; 6 Applications; 6.1 Estimating Persistence Using Multiple Subsamples; 6.2 Shape Comparison; 7 Future Work; References; A Complete Characterization of the One-Dimensional Intrinsic Čech Persistence Diagrams for Metric Graphs; 1 Introduction; 2 Background; 2.1 Homology; 2.2 Persistent Homology and Metric Graphs; 3 From Graphs to Intrinsic Čech Complexes; 3.1 Overview and Relevant Notations
3.2 Relating Graphs to Intrinsic Čech Complexes4 Proof of Main Theorem; 5 Future Work; References; Comparing Directed and Weighted Road Maps; 1 Introduction; 2 Related Work; 2.1 Weighted and Directed Graph Comparison; 2.2 Persistence-Based Comparisons; 3 Filtrations; 3.1 Swatch Filtration; 3.2 Average Minimum Distance; 3.3 Killing Cycles; 4 Comparing Data on Graphs; 4.1 Paired Analysis; 4.2 Birth-Birth Diagrams; 4.3 Examples; 5 Discussion; References; Sweeping Costs of Planar Domains; 1 Introduction; 2 Related Work; 3 Preliminaries and Notation; 3.1 Jordan Domains and Geodesics
3.2 Fréchet and Geodesic Fréchet Distances3.3 Sensor Curves; 4 Properties of Sensor Sweeps; 5 A Lower Bound on the Sweeping Cost; 6 A Lemma of No Progress; 7 Sweeping Cost of a Jordan Domain; 8 Sweeping Cost of a Convex Domain; 9 Extremal Shapes; 10 Conclusion; Appendix: Additional Lemmas and Proofs; References; Scaffoldings and Spines: Organizing High-Dimensional Data Using Cover Trees, Local Principal Component Analysis, and Persistent Homology; 1 Introduction; 1.1 Outline of Method and Paper; 1.2 Related Work; 2 Background; 2.1 Graph Theory; 2.2 Cover Tree; 2.3 Stratified Spaces
2.4 Local Principal Component Analysis2.5 Persistent Homology; 3 The Adaptive Cover Tree and the Scaffolding; 3.1 Eigenmetric Threshold; 3.2 Building the Scaffolding; 4 Local Dimension Estimation and Strata Determination; 4.1 Local Dimension Estimation; 4.2 Topology of the Strata; 5 The Spine; 6 Experiments; 6.1 Synthetic Examples; 6.2 Two Experiments on Music Structure Visualization; 7 Discussion; References; Density of Local Maxima of the Distance Function to a Set of Points in the Plane; 1 Introduction; 2 Main Result; 3 Construction; 4 Discussion and Open Problems; References