Lecture "The Multi-Cover Persistence of Euclidean Balls"

Professor Herbert Edelsbrunner is a Sackler Distinguished Lecturer in Pure Mathematics 2017/2018. 

Abstract

Given a locally finite set X in R^d and a positive radius r, the k-fold cover of X consists of all points that have k or more points of X within distance r. The order-k Voronoi diagram decomposes the k-fold cover into convex regions, and we use the dual of this decomposition to compute homology and persistence in scale and in depth.

The persistence in depth is interesting from a geometric as well as algorithmic viewpoint. The main tool in understanding its structure  is a rhomboid tiling in R^{d+1} that combines the duals for all values of k into one. We mention a straightforward consequence, namely that the cells in the dual are generically not simplicial, unless k=1 or d=1,2.

Joint work with Georg Osang.