Anna Schenfisch

The persistent homology transform, Betti function transform, and Euler characteristic transform represent a shape with a multiset of persistence diagrams, Betti functions, or Euler characteristic functions, respectively, parameterized by the sphere of directions in the ambient space. In this work, we give the first explicit construction of finite sets of directions discretizing the verbose variants of these transforms and show that such discretizations faithfully represent the underlying shape. Our discretization, while exponential in the dimension of the shape, does not depend on any restrictions on the particular immersion beyond general position, and is stable with respect to various perturbations.

Persistence modules stratify their underlying parameter space, a quality that makes persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multiparameter persistence modules. Namely, we show the K K -theory of grid multiparameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multiparameter notions of zig-zag persistence. We compare our calculations for the specific group K 0 K_0 with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.

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. Moreover, we show that this discretization is stable with respect to various perturbations and we provide an algorithm for computing the discretization. Our approach relies only on knowing the heights and dimensions of topological events, which means that it can be adapted to provide discretizations of other dimension-returning topological transforms, including the Betti function transform. With mild alterations, we also adapt our methods to faithfully discretize the Euler characteristic function transform.

Here, we introduce a new definition of regular point for piecewise-linear (PL) functions on combinatorial (PL triangulated) manifolds. This definition is given in terms of the restriction of the function to the link of the point. We show that our definition of regularity is distinct from other definitions that exist in the combinatorial topology literature. Next, we stratify the Jacobi set/critical locus of such a map as a poset stratified space. As an application, we consider the Reeb space of a PL function, stratify the Reeb space as well as the target of the function, and show that the Stein factorization is a map of stratified spaces.

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