Analysis of Matching Mass Distributions

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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.