Topological Data Analysis

Abstract

This thesis is a mathematical exposition of the theory behind Topological Data Analysis (TDA) complemented by two applications in medicine and financial realm. We start by establishing the foundation of homology theory, then study the reconstruction of the underlying manifold from point cloud data. Followed by the theory of persistent homology which provides a topological summary of the signifi cant geometrical features of the data. We study its diagram representations, robustness and characterisation via persistence modules. Subsequently, we study persistence landscapes and extend statistical concepts of confi dance intervals, convergence and hypothesis testing for topological summaries of the data. Furthermore, we discuss the mapper algorithm, which provides network representations for high dimensional data. Finally we end the thesis with a brief discussion on the interdisciplinary application of TDA implemented in this project.