Natural Stratifications of Reeb Spaces and Higher Morse Functions

Both Reeb spaces and higher Morse functions induce natural stratifications. In the former, we show that the data of the Jacobi set of a function $f:X \to \mathbb{R}^k$ induces stratifications on $X,\mathbb{R}^k$, and the associated Reeb space, and give conditions under which maps between these three spaces are stratified maps. We then extend this type of construction to the codomain of higher Morse functions, using the singular locus to induce a stratification of which sub-posets are equivalent to multi-parameter filtrations.Comment: v2: Additional examples/figures, Appendix on notions of criticality for PL ma

Efficient Graph Reconstruction and Representation Using Augmented Persistence Diagrams

Persistent homology is a tool that can be employed to summarize the shape of data by quantifying homological features. When the data is an object in $\mathbb{R}^d$, the (augmented) persistent homology transform ((A)PHT) is a family of persistence diagrams, parameterized by directions in the ambient space. A recent advance in understanding the PHT used the framework of reconstruction in order to find finite a set of directions to faithfully represent the shape, a result that is of both theoretical and practical interest. In this paper, we improve upon this result and present an improved algorithm for graph -- and, more generally one-skeleton -- reconstruction. The improvement comes in reconstructing the edges, where we use a radial binary (multi-)search. The binary search employed takes advantage of the fact that the edges can be ordered radially with respect to a reference plane, a feature unique to graphs.Comment: This work originally appeared in the 2022 proceedings of the Canadian Conference on Computational Geometry (CCCG). We have updated the proof of Theorem 2 in Appendix A for clarity and correctness. We have also corrected and clarified Section 3.2, as previously, it used slightly stricter general position assumptions than those given in Assumption

A Faithful Discretization of the Augmented Persistent Homology Transform

The persistent homology transform (PHT) represents a shape with a multiset of persistence diagrams parameterized by the sphere of directions in the ambient space. In this work, we describe a finite set of diagrams that discretize the PHT such that it faithfully represents the underlying shape. We provide a discretization that is exponential in the dimension of the shape (making it Furthermore, we provide an output-sensitive algorithm; that is, the algorithm reports the discretization in time proportional to the size of the discretization. Finally, our approach relies only on knowing the heights and dimensions of topological events, meaning that it can be adapted to provide discretizations of other dimension-returning topological transforms, including the Betti curve transform

Zig-Zag Modules: Cosheaves and K-Theory

Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig‑zag filtrations (including monotone filtrations), and for augmented persistence modules (which encode the data of instantaneous events). We then identify an equivalence of categories between a particular notion of zig‑zag modules and the combinatorial entrance path category on stratified R. Finally, we compute the algebraic K-theory of generalized zig‑zag modules and describe connections to both Euler curves and K_0 of the monoid of persistence diagrams as described by Bubenik and Elchesen

Zig-Zag Modules: Cosheaves and K-Theory

Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig‑zag filtrations (including monotone filtrations), and for augmented persistence modules (which encode the data of instantaneous events). We then identify an equivalence of categories between a particular notion of zig‑zag modules and the combinatorial entrance path category on stratified R. Finally, we compute the algebraic K-theory of generalized zig‑zag modules and describe connections to both Euler curves and K_0 of the monoid of persistence diagrams as described by Bubenik and Elchesen

Estate Planning in North Dakota: The Basics, Part 5: Gifts, Life Insurance and Annuities

Formerly published under the HE seriesFE-555 (Revised