Research in Computational Topology 2 (Association for Women in Mathematics Series, 30)

This second volume of Research in Computational Topology is a celebration and promotion of research by women in applied and computational topology, containing the proceedings of the second workshop for Women in Computational Topology (WinCompTop) as well as papers solicited from the broader WinCompTop community. The multidisciplinary and international WinCompTop workshop provided an exciting and unique opportunity for women in diverse locations and research specializations to interact extensively and collectively contribute to new and active research directions in the field. The prestigious senior researchers that signed on to head projects at the workshop are global leaders in the discipline, and two of them were authors on some of the first papers in the field. Some of the featured topics include topological data analysis of power law structure in neural data; a nerve theorem for directional graph covers; topological or homotopical invariants for directed graphs encoding connections among a network of neurons; and the issue of approximation of objects by digital grids, including precise relations between the persistent homology of dual cubical complexes. Preface The Multi-disciplinary Field of Computational Topology Working Group Project Descriptions Overview of the Contributed Chapters Final Remarks and Acknowledgments Contents The Persistent Homology of Dual Digital Image Constructions 1 Introduction 1.1 Related Work 1.2 Overview 2 Mathematical Background 2.1 Dual Cell Complexes and Filtrations 2.2 Persistent Homology 2.2.1 Definition 2.2.2 Computation 3 The Persistent Homology of Dual Filtered Complexes 4 Filtered Cell Complexes from Digital Images 4.1 Top-cell and Vertex Constructions 4.2 Modifications for Duality 5 Persistence Diagrams of the Modified Filtrations 5.1 Long Exact Sequence of a Filtered Pair 5.2 Persistence of the Image with Boundary Identified 6 Duality Results for Images 6.1 From the V-Construction to the T-Construction 6.2 From the T-Construction to the V-Construction 7 Discussion References Morse-Based Fibering of the Persistence Rank Invariant 1 Introduction 2 Notation and Definitions 3 Computing the Rank Invariant 4 Computing the Persistence Space 4.1 Restriction of a Persistence Module to Lines 4.2 Critical Values Determine the Persistence Space 4.3 Grouping Fibers of Persistence Spaces by Equivalence 5 Conclusions and Discussion Appendix: Enumerating Equivalence Classes of Lines Cuts Are Determined by Primitive Pairs Achieving Positive Slope Cuts Through a Fixed Point Retrieving Representatives Lines References Local Versus Global Distances for Zigzag and Multi-Parameter Persistence Modules 1 Introduction 1.1 Motivation 1.2 Our Contributions 2 Background and Definitions 3 Bottleneck and Wasserstein Distances in the Local vs. Global Settings 4 Applications to Metric Graphs and d-Parameter Persistence 4.1 Metric Graphs 4.2 Multi-Parameter Persistence 5 Discussion References Tile-Transitive Tilings of the Euclidean and Hyperbolic Planes by Ribbons 1 Introduction 2 Groups of Isometries and the Orbifold Fundamental Group 3 Combinatorial Tiling Theory 4 Orbifold Paths and Tile Glueing 5 Enumeration and Classification of Tile-Transitive Tilings 6 Hyperbolic Tiling Examples 7 Summary and Outlook References Graph Pseudometrics from a Topological Point of View 1 Introduction 2 Directed Graph Pseudometrics 2.1 Betti Numbers and Simplex Counts 2.2 TriadEuclid 2.3 TriadEMD 2.4 Portrait Divergence 3 Statistical Tools 3.1 Distance Correlation 3.2 Fowlkes-Mallows Index 3.3 Permutation Tests for Paired Data 3.4 k-NN Classification and Regression 4 Experimental Setup 4.1 Random Graph Models and Parameters 4.2 Computing Topological Pseudometrics dβ and d 5 Experimental Results 5.1 Fowlkes-Mallows Indices and Distance Correlation Between Pseudometrics 5.2 Classification Accuracy 5.3 Permutation Tests 5.4 Comparison of Clusterings and Classification Power 6 Conclusion References Nerve Theorems for Fixed Points of Neural Networks 1 Introduction 2 Preliminaries 2.1 Graph Theory Terminology 2.2 Background on Fixed Points of CTLNs 2.3 The DAG Decomposition 3 Directional Graphs 4 Directional Covers and Nerve Theorems 4.1 Directional Covers and Nerves 4.2 Nerve Theorems 4.3 Proofs of Nerve Theorems 5 Some Extensions and Applications 5.1 Iterating and Combining DAG Decomposition and Cycle Nerve Theorems 5.2 Extensions Beyond Directional Covers 6 Conclusion References Combinatorial Conditions for Directed Collapsing 1 Introduction 2 Background 2.1 Directed Spaces and Euclidean Cubical Complexes 2.2 Past Links of Directed Cubical Complexes 2.3 Relationship Between Past Links and Path Spaces 3 Directed Collapsing Pairs 3.1 Link-Preserving Directed Collapses 3.2 Properties of LPDCs 4 Preservation of Spaces of Dipaths 5 Discussion References Lions and Contamination, Triangular Grids, and Cheeger Constants 1 Introduction 2 Related Work 3 Notation and Definitions 3.1 Graphs 3.2 Lion Motion 3.3 Contamination Motion 3.4 Triangular Grid Graphs 4 Lions and Contamination on Triangular Grid Graphs 4.1 Sufficiency of n Lions on a Triangulated Strip 4.2 Sufficiency of "4262304 3n2"5263305 Caffeinated Lions on a Triangulated Strip 4.2.1 Sweeping Formation for "4262304 3n2"5263305 Caffeinated Lions 4.2.2 Can Lions Get to Some Predetermined Starting Position? 4.3 Insufficiency of "4262304 n2"5263305 Lions on a Triangulated Square 4.4 Conjectured Insufficiency of n22 Lions on a Triangle 5 Connection to Cheeger Constant 6 Conclusion and Open Questions References A Topological Approach for Motion Track Discrimination 1 Introduction 2 Background 2.1 Time-Delay Embedding 2.2 Topological Data Analysis 2.2.1 Persistent Homology 2.2.2 Persistence Images 3 Track Generation and Conditioning 3.1 Motion Track Generation 3.2 Motion Sub-Track Generation 4 Topological Approach for Motion Track Classification 4.1 Experimental Design 5 Results 6 Conclusion References Persistent Topology of Protein Space 1 Introduction 2 Methods 2.1 Alpha Complexes and Filtrations 2.2 Persistent Homology 2.3 Wasserstein Distances 2.4 GIT Vectors 2.5 Dimensionality Reduction 2.6 AJIVE 3 Results 3.1 Data Processing 3.2 Visualization of Protein Space 3.3 Statistical Analysis of PDs and GIT Vectors 3.4 Comparison of PDs to GIT Vectors 4 Discussion Appendix References Mappering Mecklenburg County: Exploring Census Data for Potential Communities of Interest 1 Introduction 2 Mecklenburg County, North Carolina 2.1 Data Selection 3 Methods 3.1 Background on Mapper 3.2 Mapper Inputs 4 Results 4.1 The Clusters 4.2 Hierarchical Clustering Comparison 5 Conclusions and Next Steps Appendix References Stitch Fix for Mapper and Topological Gains 1 Introduction 2 Related Work 3 Main Theoretical Result 4 Algorithm 5 Proof of Theorem 1 6 Quantifying Topological Gains 6.1 Localized Homological Difference 6.2 Local Relative Euler Characteristic 6.3 Localized Entropy Differences 7 Visualizing Topological Gains 7.1 Visualization Interface 7.2 Cylinder 7.3 Sphere 7.4 Boston Housing Dataset 7.5 Iris Dataset 7.6 Breast Cancer Wisconsin (Diagnostic) Dataset 7.7 Wine Quality Dataset 7.8 KS/NE Dataset 7.9 NKI Dataset 8 Discussion References