Optimal transport and applications in partial differential equations

In the 18th century, Gaspard Monge created a mathematical framework to find the best way to describe the optimal way to rearrange the dirt dug out from the land into castle walls or other desired shapes. More recently, in the 20th century, Leonid Kantorovich explored infinite dimensional optimization and revisited Monge's framework in order to create his own framework that improved analysis and sparked new interest in the problem. The ideas generated from studying this problem, now called optimal transport or transportation theory, have proven useful in many fields of mathematics, from PDEs to image processing and machine learning. This paper introduces the Monge and Kantorovich problems and the ideas necessary to study them. Once the optimal transportation framework is defined, this paper introduces gradient flow theory.