Rice University Department of Mathematics Colloquium

Eulerian calculations in the Wasserstein space

4:00 pm Thursday, February 15, 2018
Almut Burchard (U of Toronto)

Abstract: The optimal transport problem defines a notion of distance in the space of probability measures over a manifold, the *Wasserstein space*. In his 1994 Ph.D. thesis, McCann discovered that this space is a length space: the distance between probability measures is given by the length of minimizing geodesics called *displacement interpolants*. A number of important functionals in physics and geometry turned out to be geodesically convex; their gradient flows give rise to nonlinear degenerate diffusion equations. In contrast with classical function spaces, the Wasserstein space is not a linear space, but rather an infinite-dimensional analogue of a Riemannian manifold. In this talk, I will describe how to differentiate functionals along displacement interpolants, using an Eulerian formulation for the underlying optimal transportation problem. Time permitting, I will sketch how to justify the calculations under minimal regularity assumptions, discuss geometric implications, and mention open questions. (Joint work with Benjamin Schachter)

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