  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.
