Since the problem was formulated by G. Monge in 18C, there was not a lot of progress on the optimal transportation problem until L. Kantorovich found the relaxed problem. After L. Kantorovich, there was a lot of progress in the optimal transportation problem by many mathematicians. One of the important discoveries is that the solution of the optimal transportation problem is very closely related to a PDE called the Monge-Ampere equation. Therefore, studying some properties of the solutions to the optimal transportation problem (for example, regularity of the solution) became studying the Monge-Ampere equation.

In this talk, I will talk about the formulation of the optimal transportation problem, Kantorovich's work, and the relation to the Monge-Ampere equation. To simplify some arguments, the cost function of the optimal transportation, we will use the distance square cost function  . If time permits, I will introduce some regularity results in the Monge-Ampere equation.

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