Mathematical Sciences: The Geometry of Optimal Transportation

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Abstract McCann 9622997 This proposal focuses on the analysis of a long-standing problem which arises in economics and operations research as well as probability and statistics. Given two distributions (f and g) of mass over a Riemannian manifold M, the problem is to determine the most efficient way to rearrange the mass of the first distribution to yield the second. Efficiency is measured against a function c(x,y) which specifies the cost per unit mass for transporting material from x to y --- so the problem can be formulated as a linear program. However, when the cost function is related to the metrical distance between pairs of points on M, then the problem has a rich structure and deep connections to geometry and non-linear partial differential equations which have only begun to be explored. For costs which are either convex or concave functions of this distance, the goal would be to find a map from the manifold to itself which minimizes the total transportation costs, among maps pushing the measure f forward to g. Even on the line, this map can be intricate. Recent developments in Euclidean space suggest some promising questions to explore: What conditions guarantee existence and uniqueness of an optimal mapping? What geometrical or analytic properties characterize the optimal maps? Can this geometry be exploited fruitfully in applications? The motivation for this research is illustrated by an example from economics: Given a distribution of iron mines throughout the countryside, and a distribution of factories which require iron ore, decide which mines should supply ore to each factory in order to minimize the total transportation costs. The mines and factories lie on a curved space M, like the surface of the earth, but general enough to include barriers to transportation such as lakes and mountain ranges; the cost per ton for transporting ore from any mine to factory is determined by the distance measured between them in thi s space. Aside from industrial applications, the solution to this problem should yield new understanding of existing patterns in the economy, and may prove useful for infrastructure planning. Because of its close relationship to several areas of pure mathematics, the research promises to stimulate a fruitful exchange in two directions: powerful mathematics will be brought to bear on problems from the real word, while concrete solutions to those problems should provide new insight into the mathematics.