Curvature Sets Over Persistence Diagrams

Abstract: We study an invariant of compact metric spaces inspired by the Curvature Sets defined by Gromov. The (n,k)-Persistence Set of X is the collection of k-dimensional VR persistence diagrams of any subset of X with n or less points. This research seeks to provide a cheaper persistence-like invariant for metric spaces, as the computation of the VR complex becomes prohibitive once the input reaches a certain size. I'll focus on the case n=2k+2, where we can find a geometric formula to calculate the VR persistence diagram of a space with n points. We explore the application of this formula to the characterization of persistence sets of several spaces, including circles, higher dimensional spheres, and surfaces with constant curvature. We also show that persistence sets can detect the homotopy type of a certain family of graphs.

algebraic topologygeneral mathematicsgeneral topologygeometric topologymetric geometry

Audience: researchers in the topic

Topology, Geometry, & Data Analysis (TGDA) Seminar