Mathematical Sciences: The Geometry of Optimal Transportation

摘要麦肯9622997 这项建议的重点是分析一个长期存在的问题， 出现在经济学和运筹学以及概率论中， 统计给定黎曼流形上的两个质量分布（f和g）， 流形M，问题是确定最有效的方法来重新排列 第一次分配的质量产生第二次分配。效率 相对于函数c（x，y）进行测量，该函数指定了每单位质量的成本 把物质从x运送到y，所以这个问题可以用公式表示 作为线性规划。然而，当成本函数与 M上的点对之间的度量距离，则问题有一个 丰富的结构和深刻的联系，几何和非线性部分 微分方程才刚刚开始探索。讼费 它们是这个距离的凸函数或凹函数， 将是找到一个映射从流形本身最小化 总运输成本，在推进措施f的地图中 与g即使在线路上，这张地图也可能是错综复杂的。最近的事态发展 提出了一些有希望探索的问题：什么 条件保证存在和唯一的最佳映射？ 最优映射具有什么几何或分析性质？ 这种几何形状能在应用中得到富有成效的利用吗？ 本研究的动机是由一个例子说明， 经济：鉴于铁矿分布在整个农村， 以及需要铁矿石的工厂的分布， 我应该向每个工厂供应矿石，以使总量最小化 运输费用。矿山和工厂位于弯曲的空间M上， 就像地球的表面，但足够普遍，包括障碍， 运输，如湖泊和山脉;每吨成本为 将矿石从任何矿山运输到工厂取决于距离 在这个空间中测量它们之间的距离。 除了工业应用， 解决这一问题应使人们对现有 经济模式，并可能证明对基础设施规划有用。 由于它与纯数学的几个领域有着密切的关系， 研究工作可望在两个方向促进富有成果的交流： 强大的数学将被用来解决来自真实的问题 虽然这些问题的具体解决办法应提供新的见解， 融入到数学中

Abstract McCann 9622997 This proposal focuses on the analysis of a long-standing problem which arises in economics and operations research as well as probability and statistics. Given two distributions (f and g) of mass over a Riemannian manifold M, the problem is to determine the most efficient way to rearrange the mass of the first distribution to yield the second. Efficiency is measured against a function c(x,y) which specifies the cost per unit mass for transporting material from x to y --- so the problem can be formulated as a linear program. However, when the cost function is related to the metrical distance between pairs of points on M, then the problem has a rich structure and deep connections to geometry and non-linear partial differential equations which have only begun to be explored. For costs which are either convex or concave functions of this distance, the goal would be to find a map from the manifold to itself which minimizes the total transportation costs, among maps pushing the measure f forward to g. Even on the line, this map can be intricate. Recent developments in Euclidean space suggest some promising questions to explore: What conditions guarantee existence and uniqueness of an optimal mapping? What geometrical or analytic properties characterize the optimal maps? Can this geometry be exploited fruitfully in applications? The motivation for this research is illustrated by an example from economics: 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. The mines and factories lie on a curved space M, like the surface of the earth, but general enough to include barriers to transportation such as lakes and mountain ranges; the cost per ton for transporting ore from any mine to factory is determined by the distance measured between them in thi s space. Aside from industrial applications, the solution to this problem should yield new understanding of existing patterns in the economy, and may prove useful for infrastructure planning. Because of its close relationship to several areas of pure mathematics, the research promises to stimulate a fruitful exchange in two directions: powerful mathematics will be brought to bear on problems from the real word, while concrete solutions to those problems should provide new insight into the mathematics.