Columbia Mathematics Department Colloquium

 

"Extremal doubly stochastic measures and optimal transportation"

by

 Robert McCann  

University of Toronto

 

Abstract:
 Given a distribution of mines producing iron ore, and a distribution of factories consuming iron ore,  and knowing

the cost per ton of ore transported from a mine at X to a factory at Y, Monge's transportation problem (1781) is to

determine which mines should supply which factories in order to minimize total transportation costs. Surprisingly

 little is understood about this problem when the distributions of production and consumption live on manifolds,

and optimality is measured against a smooth cost function on the product space. I shall present a uniqueness criterion

subsuming all previous criteria, yet which is among the very first to apply to smooth costs on compact manifolds, and only

               then when the topology is simple. This new result is based on a characterization of the support of an extremal double stochastic measure.

                                            As time permits, I shall also review my surprising discovery with Young-Heon Kim (University of British Columbia and IAS)

                                         that the regularity theory of Ma, Trudinger, Wang and Loeper for optimal maps is based on a hidden pseudo-Riemannian structure,

                                             which leads to a simple direct proof of a key result in the theory, and opens several new research directions.

 

  February 18th, Wednesday, 5:00-6:00 pm

Mathematics 312

Tea will be served at 4:30pm