Analysis and geometry of optimal transportation
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:
- A. FIGALLI, Y. KIM and R.J. MCCANN, Hoelder continuity and injectivity of optimal maps (submitted July, 2011) http://front.math.ucdavis.edu/1107.1014
- A. FIGALLI, Y. KIM and R.J. MCCANN, Regularity of optimal transport maps on multiple products of spheres. (To appear in the Journal of the European Mathematical Society [JEMS]). http://front.math.ucdavis.edu/0912.3033
- A. FIGALLI, Y. KIM and R.J. MCCANN, When is multidimensional screening a convex program? Journal of Economic Theory, 146 (2011), 454-478
- Y. KIM, R.J. MCCANN and M. WARREN, Pseudo-Riemannian geometry calibrates optimal transportation.Mathematical Research Letters, 17 (2010), 1183-1197
- Y. KIM, Counterexamples to continuity of optimal transportation on positively curved Riemannian manifoldsInternational Mathematics Research Notices, 2008 (2008), article ID rnn120, 15 pages, doi:10.1093/imrn/rnn120.
- Y. KIM and R.J. MCCANN, Continuity, curvature, and the general covariance of optimal transportation Journal of the European Mathematical Society (JEMS), 12 (2010), 1009-1040