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Title: Summaries and Distances for Topological Data Analysis

Abstract: Topological data analysis starts with encoding data as a diagram of topological spaces. To this diagram we apply standard topological, algebraic, and combinatorial tools to produce a summary of our data. Next, we would like to be able perform a quantitative analysis. I will show how a family of distances, called Wasserstein distances, arise in a natural way, and how an extension of these ideas produces a summary for which we have not only distances but also angles (in fact, a Hilbert space) allowing us to apply statistics and machine learning. Key mathematical ingredients will include optimal transport, Mobius inversion, and Morse theory.

Bio: Peter Bubenik obtained his PhD from the University of Toronto in 2003 and is Professor in the Department of Mathematics and the University of Florida. He is a researcher in the NSF-Simons Southeast Center for Mathematics and Biology and he was the founding director of the Applied Algebraic Topology Research Network. In his research, he develops new tools for summarizing and visualizing large, complex, high-dimensional data by combining ideas from topology, algebra, statistics and machine learning, and he works with collaborators to use these methods to analyze data.

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