Quebec Mathematical Sciences Colloquium

September 14, 2012 from 16:00 to 18:00 (Montreal/Miami time) On location

A glimpse at the differential topology and geometry of optimal transportation

Colloquium presented by Robert McCann (University of Toronto)

The Monge-Kantorovich optimal transportation problem is to pair producers with consumers so as to minimize a given transportation cost. When the producers and consumers are modeled by probability densities on two given manifolds or subdomains, it is interesting to try to understand the structure of the optimal pairing as a subset of the product manifold. This subset may or may not be the graph of a map. The talk will expose the differential topology and geometry underlying many basic phenomena in optimal transportation. It surveys questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.

Address

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., room SH-3420