Vlad Kobzar
The Ohio State University
Title
Online Komlos converges to mean curvature flow
Abstract
We establish a direct connection between combinatorial discrepancy minimization problems and curvature flows in R^m. This is done by determining the long-time asymptotics of an online version of the classic vector balancing problem, known as the Komlós conjecture, and showing it is exactly determined by the extinction time of mean curvature flow on the m-dimensional cube. Our proof builds upon Kohn and Serfaty's work on deterministic games and mean curvature flow, and Banaszczyk's Euclidean analogue of the Beck-Fiala theorem. As a consequence of this geometric characterization, we show that the leading order term of the value of this game grows as \Theta (\sqrt { T \log m}) as the time horizon T gets large. This is joint work with Nestor Guillen at Courant. A geometric PDE-inspired perspective is also fruitful in the more classic, offline Komlós setting, and I will briefly preview our most recent results in this direction at the end of the talk.
