In this series of lectures we introduce the Monge-Kantorovich problem ofoptimally transporting one distribution of mass onto another, where optimalityis measured against a cost function c(x,y). Connections to geometry,inequalities, and partial differential equations will be discussed, focusing inparticular on recent developments in the regularity theory for Monge-Amperetype equations. An application to microeconomics will also be described, whichamounts to finding the equilibrium price distribution for a monopolistmarketing a multidimensional line of products to a population of anonymousagents whose preferences are known only statistically.
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