Optimal transport: from two to many marginals

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This proposal revolves around optimal transport, the variational problem of coupling two probability measures (marginals) in order to minimize the expected value of a prescribed cost function. The problem dates back to Gaspard Monge in 1781, who was interested in moving a pile of dirt into a hole of the same volume in order to minimize the average distance that the dirt moves (here the probability densities are given by the height and depth of the pile and hole, respectively, and the cost function is the Euclidean distance). This field has flourished since the late 80's; it has many applications, both within and beyond mathematics, and touches on analysis, partial differential equations, geometry and probability. The structure of solutions is now quite well understood; conditions under which the minimizer lies on a graph over one of the variables, and is unique, are well known. These optimal maps can be characterized by solutions to certain Monge-Ampere type partial differential equations and a deep regularity (or smoothness) theory has been developed. The particular structure of solutions plays a key role in many applications.***Much of the proposed research involves a variant known as multi-marginal optimal transport; this is the problem of aligning several probability distributions with maximal efficiency, again relative to a given cost function. Interest in multi-marginal problems is relatively new, but has increased exponentially over the past few years, due largely to a surprisingly diverse collection of emerging applications: aligning electrons to minimize interaction energy in density functional theory, interpolating between distributions in data science, matching agents in multi-sided markets in economics, etc. Though there has been significant progress, much remains to be done in order to have a complete understanding of the structure of solutions. ***An important general theme is to understand which properties of the two marginal problem carry over to the multi-marginal setting. The answer depends on the cost function in ways which are subtle and still only partially understood. A dichotomy has begun to emerge between nice cost functions, for which the solution behaves much like in the two marginal case (solutions are unique and concentrate on graphs over one of the variables) and those for which they exhibit much more exotic and unexpected behaviour (solutions may concentrate on high dimensional sets and be non-unique). The classification remains crude, and solutions on both sides must be better understood.