The Little Diagram That Could: Geometric Properties and Statistical Applications of Persistence Diagrams in Topological Data Analysis

Topological Data Analysis (TDA) is a collection of techniques for data analysis that leverages topological invariants of spaces formed from data points. These methods excel at extracting useful information from noisy or sparse data, making them attractive to many mathematicians, statisticians, and scientists. In this thesis, we explore TDA on three fronts: algebraic foundations, statistical applications, and metric properties. Throughout, the central object of study is the Persistence Diagram (PD), a summary of the changes in homology that occur as one builds simplicial complexes from the data by increasing a parameter.