Analysis of Matching Mass Distributions

拟议的研究涉及最优运输理论，该理论解决了两个或多个质量分布以最有效的方式匹配时所产生的问题，最大限度地减少了将质量从一个位置移动到另一个位置的运输成本。 一个重点将是发展数学方法，了解最佳运输地图，这可能是由于缺乏足够的几何条件的成本和质量分布的奇点的性质。这将被认为是在弯曲的景观运输。此外，大量的努力将致力于许多质量分布的情况下，要匹配。另一个目标是发展一个理论的最佳运输问题，模型的情况下，质量分布运输沿着渠道有限的能力。 这项研究将对最优输运理论产生重大影响，通过给出最优输运现象的基本性质的重要结果，这些现象在数学分析和几何，数学经济学，概率论，流体/连续介质力学，气象学和生物学等领域有许多当前和潜在的应用。

The proposed research concerns optimal transport theory which addresses problems arising when two or more mass distributions are to be matched in a most efficient way, minimizing a transportation cost of moving mass from one location to another. A focus will be on developing mathematical methods for understanding the nature of singularities of optimal transport maps, which may occur due to lack of sufficient geometrical conditions on the cost and the mass distributions. This will be considered for transportation on curved landscapes. Also, a substantial effort will be devoted to the case of many mass distributions to be matched. Another goal is to develop a theory for an optimal transport problem which models the situation where a mass distribution is transported along channels with limited capacities. This research will have significant impact on the optimal transport theory, by giving important results on the fundamental properties of optimal transport phenomena, which have many current and potential applications in mathematical analysis and geometry, mathematical economics, probability theory, fluid/continuum mechanics, meteorology, and biology, among others.