Wesley Hamilton (UNC-CH), GMA & Visions Seminar

Title:  Topological Data Analysis: A Survey

Abstract: With the advent of computers followed a boom in data sets; in practical terms, the growing sizes of data sets outpaced the growing computational power. This made it infeasible to analyze large data sets at once, so other qualitative techniques were needed. One solution was to analyze the topology of a data set. Since data sets are collections of discrete points (and hence have not very interesting topology), one must rst associate some geometric structure to the data points, which then allows us to compute topological invariants. We are thus left with three questions (from which many more will spring): what is a good structure to impose on a given data set?  How do we compute these topological invariants in practice? and what can we infer about the structure of the data from these invariants?  The goal of this talk is to give an overview of Topological Data Analysis, with an emphasis on the (pure) tools and techniques that have proven useful in analyzing large data sets. In particular we’ll look at how algebraic topology, algebraic geometry, category theory, and Morse theory all play some role in this new and exciting eld. The intended audience is everyone!