Computing the Multicover Bifiltration.

Given a finite set , let denote the set of all points within distance to at least points of . Allowing and to vary, we obtain a 2-parameter family of spaces that grow larger when increases or decreases, called the . Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy.

A Simple Algorithm for Higher-Order Delaunay Mosaics and Alpha Shapes.

We present a simple algorithm for computing higher-order Delaunay mosaics that works in Euclidean spaces of any finite dimensions. The algorithm selects the vertices of the order- mosaic from incrementally constructed lower-order mosaics and uses an algorithm for weighted first-order Delaunay mosaics as a black-box to construct the order- mosaic from its vertices. Beyond this black-box, the algorithm uses only combinatorial operations, thus facilitating easy implementation.

The Multi-Cover Persistence of Euclidean Balls.

Given a locally finite and a radius , the - of and consists of all points in that have or more points of within distance . We consider two filtrations-one in obtained by fixing and increasing , and the other in obtained by fixing and decreasing -and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth.

Topological signatures and stability of hexagonal close packing and Barlow stackings.

Two common representations of close packings of identical spheres consisting of hexagonal layers, called Barlow stackings, appear abundantly in minerals and metals. These motifs, however, occupy an identical portion of space and bear identical first-order topological signatures as measured by persistent homology. Here we present a novel method based on -fold covers that unambiguously distinguishes between these patterns.

A step in the Delaunay mosaic of order .

Given a locally finite set and an integer , we consider the function on the dual of the order- Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order- (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551-559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76-83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90-145, 1998) and Freij (Discrete Math 309:3821-3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps.