I shall briefly review new and old results concerning optimal transportation of mass between two manifolds, in particular the regularity theory of Ma, Trudinger, Wang and Loeper, and counterexamples due to Loeper. I will then describe an unexpected semi-Riemannian structure underlying these results, which yields the much-desired direct proof of a key result in this theory, and opens new directions.
This represents joint work with Young-Heon Kim (University of Toronto).
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