# UBC Mathematics Colloquium

## Robert McCann

(University of Toronto)

## Extremal Doubly Stochastic Measures and Optimal Transportation

### Fri., Nov. 13, 2009, 3:00pm, MATX 1100

Abstract:

Imagine some commodity being produced at various locations and consumed
at others. Given the cost per unit mass transported,
the optimal transportation problem is to pair consumers with producers
so as to minimize total transportation costs. Despite much
study, surprisingly little is understood about this problem
when the producers and consumers are continuously distributed over
smooth manifolds, and optimality is measured against a cost function
encoding some geometry of the product space.

This talk will be an introduction to the optimal transportation, its
relation to Birkhoff's problem of characterizing of extremality among
doubly stochastic measures, and recent progress linking the two.
It culminates in the presentation of a criterion for uniqueness of
solutions which subsumes all previous criteria, yet which is among the
very first to apply to smooth costs on compact manifolds, and
only then when the topological type of one of the two underlying
manifolds is the sphere.