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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium

Robert McCann
University of Toronto
Title: Multi- to one-dimensional transportation

Abstract: Monge and Kantorovich's problem of optimal transportation plays a central role bridging from analysis, geometry and dynamical systems to applications ranging from engineering design to weather prediction, image processing and stable marriage to optimal decision-making. It involves coupling two distributions of mass as efficiently as possible with respect to some specified cost criterion. Typically the mass distributions are given by probability densities on the same space, or on two spaces with the same finite dimension.

In this talk we consider instead the problem of transportation between spaces of unequal dimensions. If target space is the real line, we describe a nesting criterion relating the cost to the densities, under which it becomes possible to uniquely solve this problem, by constructing an optimal map one level set at a time. This map is continuous if the target density has connected support. We use level-set dynamics to develop and quantify a local regularity theory for this map and the dual potentials (which play the role of location dependent prices quantifying remoteness of supply relative to demand). We give examples which highlight obstructions to global regularity results.

Date: Tuesday, December 15, 2015
Time: 4:30 pm
Location: Wean Hall 7218
Submitted by:  Ian Tice