Department of Mathematics
Mathematics in Science & Society
by
Prof. Robert McCann

In the classical transportation problem, one is given a distribution of iron mines throughout the countryside, and a distribution of factories which reqiure iron ore, and asked to decide which mines should supply ore to each factory in order to minimize the total transportation costs. When the mines and factories are distributed continuously throughout Euclidean space - or a Riemannian manifold - and the cost per ton of ore transported from the mine at x to the factory at y is specified by a function of the distance, the problem turns out to have deep connections to geometry and non-linear PDE.

In a number of cases, the solution takes the form of a measure-preserving map between the measures which specify the distribution of mines and factories. This map is unique, and uniquely characterized by its geometry. Even on the line this mapping may be intricate. This talk explores some unexpected features of the solution for concave costs, including the emergence of a local/global hierarchy which seems as fascinating from the economic as the mathematical point of view. In one-dimension, this structure may be exploited to provide an algorithm for obtaining exact solutions to the infinite dimensional problem by a combinatorial sequence of finite dimensional optimizations involving convex, separable network flows.