Variational problems in mathematics and the sciences

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***This proposal addresses the analysis of a collection of variational ***problems with applications in mathematics and the sciences.***Central among them is an optimal transportation problem***of Monge and Kantorovich, which can be caricatured as follows:***given a collection of iron mines and a collection of factories ***consuming iron ore, decide which mine should supply***which factory in order to minimize the total transportation costs.***When the mines and factories are distributed continuously throughout***a curved landscape, with geometry reflected in the transport cost,***the problem has deep connections to***geometry, topology, dynamical systems, and nonlinear partial differential equations.***Despite much recent progress, fundamental questions remain to be resolved. ***We target these, and striking applications of this problem ***to models in economics and in physics.******The last decade has seen enormous advances in our understanding of such***issues. The answers depend heavily on***the choice of cost. Topological restrictions on the cost guarantee***that there is only one solution and every mine supplies a single factory; ***under further geometric restrictions this correspondence is smooth. ***However for smooth transportation costs on compact manifolds such as the***spherical surface of the earth one or both conditions must fail.***Characterizing discontinuities then becomes an interesting question,***The resulting transport patterns are much more complicated: we plan to investigate***uniqueness of solutions, and what can be said about dimension,***branching, and smoothness of their support. ******At the same time, we tackle variants such as: how does congestion affect the situation? ***Generically, the solution saturates the congestion bound***wherever transport occurs. Thus it is completely determined by a separation of***active from inactive transportation routes. We plan to investigate the analytic and geometric***properties of this separation. ******Finally, we target select applications to physics and to economics. The discontinuities***described above correspond to pressure fronts in the semigeostrophic theory***of atmospheric motion. We seek to analyze their formation and evolution.***Nonlinear diffusions model phenomena ranging from electronic transport***in semiconductors to the dynamics of thin films; we seek to analyze their long time***behaviour using a gradient flow formulation of the dynamics with respect to***transportation costs. Stable marriage, optimal-pricing and team-building***problems also involve the solution of continuous linear programs.***We seek an analysis of the models which improves our understanding***of the underlying natural phenomena, and leads to better decision-making principles***concerning allocation of resources, matching problems, team-building, and***the ability to influence, explain, exploit, and control economic phenomena.**