Five lectures on optimal transportation: Geometry, regularity and applications

TL;DR

This paper provides an overview of optimal transportation theory, exploring its geometric, analytical, and economic applications, with a focus on recent advances in regularity theory for Monge-Ampère equations.

Contribution

It introduces recent developments in the regularity theory of Monge-Ampère equations and connects optimal transportation to geometry, inequalities, PDEs, and microeconomics.

Findings

Advances in regularity results for Monge-Ampère equations.

Connections established between optimal transport and geometric inequalities.

Application demonstrated in economic modeling of equilibrium prices.

Abstract

In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Ampere type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.