## PIMS/UBC Distinguished Lecture: Robert McCann

- Date: 08/20/2012
- Time: 15:00

**Robert McCann**

Robert McCann has been a Professor of Mathematics at the University of Toronto since 1998.

He grew up in Windsor Ontario, and studied engineering and physics at Queen's University before completing his doctorate in mathematics at Princeton in 1994. In the subsequent years, McCann has emerged as a leading figure in the development of the theory of optimal transportation. His work balances very pure contributions to deep mathematics with the discovery of new applications to image processing, atmospheric circulation patterns, and to optimizing economic decisions.

Before accepting a position at the University of Toronto, McCann was Tamarkin Assistant Professor of Mathematics at Brown University in Providence, Rhode Island.

His past awards include an American Mathematical Society Centennial Fellowship, the Monroe Martin Prize in Applied Mathematics, and the Coxeter-James Prize of the Canadian Mathematical Society. He currently serves as Editor-in-Chief of the Canadian Journal of Mathematics.

University of British Columbia

A glimpse into the differential geometry and topology of optimal transportation.

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.

See preprint at www.math.toronto.edu/mccann/publications

Location: WMAX 110