Analysis of Matching Mass Distributions

拟议的研究涉及最优运输理论，该理论解决了当两个或更多质量分布要以最有效的方式匹配时产生的问题，从而最小化将质量从一个位置移动到另一个位置的运输成本。 重点将是开发数学方法，以了解最优运输地图奇点的性质，这种奇点可能是由于缺乏关于成本和质量分布的充分几何条件而发生的。这将被考虑用于在弯曲的景观上进行运输。此外，将在许多质量分布相匹配的情况下投入大量的努力。另一个目标是发展一种关于最优运输问题的理论，该问题模拟沿有限能力的航道运输质量分布的情况。 这项研究将对最优传输理论产生重大影响，因为它给出了关于最优传输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具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.