Title: Using Topology to Measure Shape in Data
Abstract: Data of various kinds is being collected at an enormous rate, and in many different forms. Often, the data are equipped with a notion of distance that reflects similarity in some sense. Using this similarity measure, certain topological features–e.g. the number of connected components, loops, and trapped volumes–can be ascertained and can provide insight into the structure of these complex data sets. In this talk, I will introduce topology and a fundamental tool of topological data analysis, persistent homology. Then, we will see how these tools can be used for clustering, with machine learning, and to explain features in data. In particular, we will discuss (1) using persistence to explore the relationship between country development and geography, (2) vectorizing persistence information via a persistence image to analyze the discrete dynamical system of the linked twist map, and (3) explore notions of minimal generators to extract geometric meaning from homological features.
Dr. Ziegelmeier is an Associate Professor at Macalester College.