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**