Despite years of intense study, surprisingly little is understood about the optimal transportation of a mass distribution from one manifold to another, where optimality is measured against a cost function on the product space.
I shall present a uniqueness criterion subsuming all previous criteria, yet which is among the first to apply to smooth costs on compact manifolds, and only then when the topology is simple. I shall review the regularity theory of Ma, Trudinger, Wang and Loeper for optimal maps, and the counterexamples of Loeper, before explaining my surprising discovery with Young-Heon Kim (University of Toronto) that this theory is based on a hidden semi-Riemannian structure, which yields the desired direct proof of a key result in the theory, and opens many new research directions.
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