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My research interests include algebraic geometry, representation theory, and combinatorics, as well as applied topology—currently with little overlap between these areas. I'm happy to work with students in any of these fields, but this talk will focus on applied topology.

I'll describe a framework where transport maps—familiar from optimal transport theory and the Monge problem—emerge naturally as a kind of homotopy. This perspective has direct applications in topological data analysis, where loss functions often encode the structure of data. I’ll describe other implications in a more general context of statistical models, and summarize some of the specific applications I am interested in.