报告摘要: The optimal transportation problem was first introduced by Monge in 1781. By Kantorovich's duality, this problem can be formulated as a Monge-Ampere type equation, and the existence and regularity of solutions have been obtained under certain conditions. The cost function in Monge's original problem is at the borderline of these conditions. With my collaborators Qi-Rui Li and Filippo Santambrogio, we recently studied the regularity of Monge's problem and discovered some delicate results. We proved that in a smooth approximation, the eigenvalues of the Jacobian matrix of the optimal mapping are uniformly bounded but the mapping itself may not be Holder continuous. But in dimension two the mapping is continuous. In this talk I will discuss recent development in this direction.