UBC Mathematics Colloquium

Robert McCann
(University of Toronto)


Extremal Doubly Stochastic Measures and Optimal Transportation

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


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.

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