Analyzing Stratified Spaces Using Persistent Versions of Intersection and Local Homology

Abstract

This dissertation places intersection homology and local homology within the framework of persistence, which was originally developed for ordinary homology by Edelsbrunner, Letscher, and Zomorodian. The eventual goal, begun but not completed here, is to provide analytical tools for the study of embedded stratified spaces, as well as for high-dimensional and possibly noisy datasets for which the number of degrees of freedom may vary across the parameter space. Specifically, we create a theory of persistent intersection homology for a filtered stratified space and prove several structural theorems about the pair groups asso- ciated to such a filtration. We prove the correctness of a cubic algorithm which computes these pair groups in a simplicial setting. We also define a series of intersec- tion homology elevation functions for an embedded stratified space and characterize their local maxima in dimension one. In addition, we develop a theory of persistence for a multi-scale analogue of the local homology groups of a stratified space at a point. This takes the form of a series of local homology vineyards which allow one to assess the homological structure within a one-parameter family of neighborhoods of the point. Under the assumption of dense sampling, we prove the correctness of this assessment at a variety of radius scales.