Speaker： XIA Qinglan, University of California at Davis
Time：2016-03-18 16:00-17:00
Place：1218, School of Mathematical Sciences

Detail：The optimal transport problem aims at finding an optimal wayto transport a given probability measure into another. In contrast tothe well-known Monge-Kantorovich problem, the ramified optimaltransportation problem aims at modeling a branching transport networkby an optimal transport path between two given probability measures.Such an optimal transport path may be viewed as a geodesic in the spaceof probability measures under a suitable metric. This geodesic mayexhibit a tree-shaped branching structure found in many applicationssuch as trees, blood vessels, draining and irrigation systems. Here,we extend the study of ramified optimal transportation between probabilitymeasures from Euclidean spaces to a geodesic metric space. We investigatethe existence as well as the behavior of optimal transport paths undervarious properties of the metric such as completeness, doubling, orcurvature upper boundedness. We also introduce the transport dimension ofa probability measure on a complete geodesic metric space, and show thatthe transport dimension of a probability measure is bounded above by theMinkowski dimension and below by the Hausdorff dimension of the measure.Moreover, we introduce a metric, called "the dimensional distance", onthe space of probability measures. This metric gives a geometric meaningto the transport dimension: with respect to this metric, the transportdimension of a probability measure equals to the distance from it to anyfinite atomic probability measure.

Organizer: School of Mathematical Sciences