Optimal transportation: geometry and dynamics

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This proposal focuses on the analysis of a family of optimization and evolution problems having applications in physics, geometry, medical imaging, partial differential equations, and economics. Linking them together is a transportation problem --- which comes into play whenever the dynamics conserve a local quantity. It can be caricatured as follows: 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. Here the cost per ton of ore transported from the mine at x to factory at y is specified by a function c(x,y). However, when the mines and factories are distributed continuously over a curved landscape, then the problem has a rich structure and deep connections to geometry, nonlinear partial differential equations and the topology of the cost function, which have only begun to be explored.