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Number Theory Seminar

Number Theory and Optimal Transport: an unusual connection

Stefan Steinerberger (Yale)

Location:  Hill 525
Date & time: Tuesday, 08 October 2019 at 2:00PM - 3:00PM

Abstract:  Optimal Transport studies the problem of how to move one measure to another so that the "transport cost" is minimal. Think of one measure being products in a warehouse and the other measure being how much people want to buy the product: the transport distance would then be the amount of miles trucks have to drive (weighted by how much they carry). I will start by giving a gentle Introduction to this topic, we do not actually need very much. My question then is: suppose one measure is the normalized counting measure in quadratic residues in a finite field and the other is the uniform measure, can the Transport be estimated? Or maybe Dirac measures placed in irrational rotations on the Torus: how cheap is it to transport them to the Lebesgue measure? And are these results interesting? (Spoiler: yes). Some recent advances in Optimal Transport allow these problems to be reduced to a simple exponential sum. There are many, many open questions.

I will discuss two fun byproducts in particular: (1) improving a standard estimate in numerical integration theory from the 1950s that was considered to be sharp (with a very simple transport argument) and (2) a mysterious new phenomenon involving exponential sums that arose along the way (it can be formulated simply in terms of polynomials but that doesn’t seem to make it any easier).