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.