Optimal transportation: geometry and dynamics

这项建议侧重于分析一类在物理、几何、医学成像、偏微分方程式和经济学中有应用的优化和进化问题。将它们连接在一起是一个交通问题-只要动力保持当地的数量，这个问题就会发挥作用。它可以被讽刺地描述为：鉴于铁矿在农村的分布，以及需要铁矿石的工厂的分布，决定哪些矿山应该向每个工厂供应铁矿石，以将总运输成本降至最低。这里，从x处的矿山运输到y处的工厂的每吨矿石的成本由函数c(x，y)指定。然而，当矿山和工厂连续分布在弯曲的地貌上时，该问题具有丰富的结构和与几何、非线性偏微分方程组和代价函数的拓扑的深刻联系，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.