Mathematical Sciences: The Geometry of Optimal Transportation

摘要麦肯9622997这项建议侧重于分析经济学和运筹学以及概率和统计学中出现的一个长期存在的问题。给定黎曼流形M上的两个质量分布(f和g)，问题是确定重新排列第一个分布的质量以产生第二个分布的最有效方法。效率是根据函数c(x，y)来衡量的，该函数规定了从x到y运输材料的单位质量成本-因此问题可以用线性规划来表示。然而，当代价函数与M上的点对之间的度量距离有关时，该问题具有丰富的结构和与几何和非线性偏微分方程组的深刻联系，这些问题才刚刚开始探索。对于这种距离的凸函数或凹函数的成本，目标是找到一个从流形到流形本身的映射，在将度量f向前推到g的映射中，使总的运输成本最小。即使在生产线上，该映射也可能是复杂的。欧几里德空间的最新发展提出了一些值得探讨的问题：什么条件保证了最优映射的存在和唯一性？最优地图的几何或解析性质是什么？这种几何能否在应用中得到丰硕的应用？这项研究的动机从经济学的一个例子中得到了说明：考虑到铁矿在农村的分布，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。矿山和工厂位于一个弯曲的空间M上，就像地球表面一样，但普遍到足以包括湖泊和山脉等运输障碍；从任何矿山到工厂的每吨矿石运输成本取决于它们在S空间中测量的距离。除了工业应用，这个问题的解决方案应该会对经济中现有的模式产生新的理解，并可能被证明对基础设施规划有用。由于它与纯数学的几个领域密切相关，这项研究有望在两个方向上促进富有成效的交流：强大的数学将应用于现实世界中的问题，而这些问题的具体解决方案应该为数学提供新的见解。

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.