Multi-marginal optimal transportation and applications.

这一建议围绕多边际最优运输问题解的几何和光滑性展开。这就是相对于给定的盈余函数尽可能有效地对齐几个规定的质量分布或边际的问题。为了帮助说明这个问题，假设一家大型建筑公司雇佣了木匠、电工和水管工来建造大量的房屋。因此，公司需要组建团队；团队由每种类型的行业人员组成，负责建造一套房子。当然，不同的员工组合或多或少会提高效率，而构建团队以使整个流程尽可能高效是符合公司最佳利益的。我的研究重点是这种最优排列的结构。例如，每个木匠只与一个电工和一个管道工一起工作是最好的，还是让一些木匠在几个不同的团队中兼职工作更好？在第一种情况下，分配给他们工作的木匠的电工工作顺利吗？当每个团队的规模(即工人类型的数量)变得非常大时会发生什么？令人惊讶的是，这类问题与偏微分方程、几何和概率等数学领域相互作用很深，在经济学、数学金融、统计物理和图像处理中有许多潜在的应用。然而，尽管在只有两种类型的工人的类似问题上取得了很大进展，但在团队由三种或更多类型的工匠组成的情况下，人们知道的相对较少。这项研究的主要目标是促进科学知识的进步；然而，在上述领域出现了许多有前途的、潜在的应用。

This proposal revolves around the geometry and smoothness of solutions to multi-marginal optimal transportation problems. This is the problem of aligning several prescribed mass distributions, or marginals, as efficiently as possible, relative to a given surplus function. To help illustrate the problem, suppose a large construction company employs, for example, carpenters, electricians and plumbers in order to build a large number of houses. The company, then, needs to make teams; a team consists of one of each type of trades person and will be responsible for building one house. Of course, different combinations of workers will be more or less efficient and it is in the company's best interests to construct its teams such that the overall process is as efficient as possible. My research focuses on the structure of this optimal alignment. Is it optimal, for example, for each carpenter to work with exactly one electrician and one plumber, or is it better for some carpenters to work part-time on several different teams? In the first case, is the assignment of, say, electricians to the carpenters they work with smooth? What happens as the size of each team (ie, the number of types of workers) becomes very large? Surprising, problems of this type interact deeply with mathematical fields such as partial differential equations, geometry and probability and have numerous potential applications in economics, mathematical finance, statistical physics and image processing. However, despite great progress on similar problems with only two types of workers, relatively little is known in the case where teams consist of three or more types of tradespeople. The primary goal of this research is the advancement of scientific knowledge; however, there are many promising, potential applications emerging in the areas mentioned above.