Topological Data Analysis for Shape Comparison

Abstract

The goal of the field of topological data analysis (TDA) is to quantitatively encode and measure shape in data using Algebraic Topology. The available tools encompass both algebraic constructions (such as persistence diagrams and Euler characteristics) as well as graph based representations (such as Reeb graphs, mapper graphs, and merge trees). Applications of TDA have exploded in recent years, finding use in a diverse array of domains including plant biology, neuroscience, mechanical engineering, and many more. This increased interest is due to its now extensive theoretical foundation, and more recently due to the increased availability of more efficient algorithms and software making TDA pipelines more readily accessible to domain scientists. In this talk, we will review some of the tools available with a particular focus on encoding embedded shapes in d- dimensional Euclidean space (with most of our applications living in the setting of d=2 or 3), and for creating metrics between these representations to allow for access to tools such as statistics and machine learning.

Elizabeth Munch is an Associate Professor at Michigan State University with a primary appointment in the Department of Computational Mathematics, Science, and Engineering and a secondary appointment in the Department of Mathematics. Liz earned her PhD from the Department of Mathematics at Duke University in May 2013. Prior to joining the faculty of MSU, she was an Assistant Professor in the Department of Mathematics and Statistics at the University at Albany - SUNY, and a Postdoctoral Fellow at the Institute for Mathematics and its Applications at the University of Minnesota.