Optimal transportation: geometry and dynamics

本建议着重分析在物理、几何、医学成像、偏微分方程和经济学中应用的一系列优化和进化问题。将它们连接在一起是一个运输问题——每当动态保持局部数量时，这个问题就会发挥作用。它可以被讽刺如下：给定整个农村的铁矿分布，以及需要铁矿石的工厂分布，决定哪个矿山应该向每个工厂供应矿石，以使总运输成本最小化。这里每吨矿石从x点的矿山运输到y点的工厂的成本用函数c（x,y）表示。然而，当矿山和工厂连续分布在一个弯曲的景观上时，这个问题就具有丰富的结构，并且与几何、非线性偏微分方程和成本函数的拓扑有很深的联系，而这些才刚刚开始探索。

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.