A STUDY OF PERSISTENT HOMOLOGY IN TOPOLOGICAL DATA ANALYSIS

Abstract

This thesis studies how persistent homology can recover stable topological features from high-dimensional noisy data. It describes a mathematical setup with homology theory and simplicial complexes (like Rips and Cech complexes) where multi-scale data representations can be built. By building persistence modules and applying algebraic techniques over polynomial rings, the thesis highlights efficient algorithms for the computation of the lifetime of features like connected components and loops, with both theoretical arguments and the usage of computational software for topological data analysis.