What is Topological Data Analysis?

Abstract: This talks serves as an introduction to Topological Data Analysis (TDA) – a method for analyzing the shape and structure of data using concepts from topology. TDA merges algebraic topology and other mathematical tools with real-world statistical analysis, offering both elegant mathematical theory and practical applications. The focus of this talk will be on Persistent Homology, widely regarded as the cornerstone technique in TDA. We’ll begin by exploring “why should we care about the topology or shape of data, and in what contexts is TDA valuable?” through examples such as brain artery tree data and prostate cancer histopathology images. Then, we’ll provide a gentle introduction to key concepts in homology theory (within algebraic topology) and demonstrate how these ideas evolve into the concept of persistent homology. Finally, we will examine how we could perform statistical analysis on persistent homology - through vectorization.