Analysis and geometry of optimal transportation

Associated People:

Young-Heon Kim (University of British Columbia)

Associated Sites:
PIMS University of British Columbia

In optimal transport theory, one wants to understand the phenomena arising when a mass distribution is transported to another in a most efficient way, where efficiency is measured by a given transportation cost function. For example, consider the problem of how to match water resources and towns that are distributed over a region.

A fundamental mathematical issue is whether such matching, called an optimal map, is continuous. For example, if two towns are located close to each other, are they matched to water resources that are also nearby?  It is not the case in general and deep mathematical structures are involved. In particular, such a problem is reduced to studying solutions to a fully nonlinear partial differential equation of Monge-Ampére type and it is related to the geometry of the domain (i.e. the landscape) and the transportation cost.  Kim and his collaborators have proven various continuity results for optimal maps under a sharp condition, now called Ma-Trudinger-Wang condition, and also found unexpected connections to symplectic and pseudo Riemannian geometry as well as microeconomics problems. 

Some examples of Kim’s related work include: