Persistent homology : categorical structural theorem and stability through representations of quivers

Abstract

The purpose of this thesis is to advance the study and application of the field of persistent homology through both categorical and quiver theoretic viewpoints. While persistent homology has its roots in these topics, there is a wealth of material that can still be offered up by using these familiar lenses at new angles. There are three chapters of results. Chapter 3 discusses a categorical framework for persistent homology that circumvents quiver theoretic structure, both in practice and in theory, by means of viewing the process as factored through a quotient category. In this chapter, the widely used persistent homology algorithm collectively known as reduction is presented in terms of a matrix factorization result. The remaining results rest on a quiver theoretic approach. Chapter 4 focuses on an algebraic stability theorem for generalized persistence modules for a certain class of finite posets. Both the class of posets and their discretized nature are what make the results unique, while the structure is taken with inspiration from the work of Ulrich Bauer and Michael Lesnick. Chapter 5 deals with taking directed limits of posets and the subsequent expansion of categories to show that the discretized work in the second section recovers classical results over the continuum.