Analysis of Matching Mass Distributions

所提出的研究涉及最优运输理论，该理论解决了当两个或多个质量分布以最有效的方式匹配时出现的问题，使质量从一个地方移动到另一个地方的运输成本最小化。重点将放在发展数学方法，以了解最优运输图的奇点性质，这种奇点可能由于缺乏关于成本和质量分布的足够几何条件而发生。这将被考虑在弯曲景观上的交通。此外，对于许多需要匹配的质量分布的情况，将投入大量的努力。另一个目标是发展最优运输问题的理论，该理论模拟了质量分布沿着有限容量的渠道运输的情况。本研究将对最优输运理论产生重大影响，对最优输运现象的基本性质给出重要结果，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.