Analytic Number Theory and Optimal Transport: an interesting connection

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). And do they carry some useful meaning? (Spoiler: yes) Some recent advances in Optimal Transport allow these problems to be reduced to a simple exponential sum; basic ingredients from Analytic Number Theory can then be used to get new insight at relatively low technical cost. There are many, many open questions.

number theory

Audience: researchers in the topic

Rutgers Number Theory Seminar

Series comments: Seminar talks in this series will be conducted via Zoom. To join our mailing list and/or receive Zoom links and meeting passwords, please email Alexander Walker (alexander.walker@rutgers.edu).