Persistent Homology in Topological Data Analysis: Theory and Applications

Abstract

Persistent homology is a computational methodology for extracting topological features of data across a range of scales. Originating from the work of Edelsbrunner, Letscher and Zomorodian in 2002, it has developed into a mature branch of topological data analysis with applications in neuroscience, materials science, genomics, and machine learning. This paper reviews the mathematical foundations of persistent homology, describes its computational realisation via simplicial filtrations and persistence diagrams, and surveys representative applications. Stability theorems and the bottleneck distance provide a principled metric on persistence diagrams, enabling statistical inference and integration with machine-learning pipelines. Current research directions include multi-parameter persistence, differentiable persistence modules, and scalable algorithms for large data.