Regular Location: THO 217
We introduce formal and rigorous ideas from the field of optimal transport. This is the first of three talks, leading up to a simple application of mass transport in the context of an inverse problem.
In this first talk we'll give an advertisement for the tools of optimal transportation, highlighting their contribution to diffusion equations, simple proofs of Sobolev and isoperimetric inequalities, generalizing the Ricci-bounded-below condition beyond smooth manifolds, and geometrically reinterpreting the Schroedinger equation. We'll then learn about two ideas at the center of these applications: 1) that probability measures can be formally seen as a Riemannian manifold (F. Otto '01) and 2) certain entropy functionals are convex in this geometry (R. McCann '94). We'll fill out the hour by reviewing the formal Riemannian structure (local geometry) and rigorous aspects of (global) Wasserstein distance.
We'll see how the probability measures on $\R$ have a particularly simple (flat) geometry, in which the convexity of entropy is easily calculated. We'll do a convexity calculation for a class of entropies on a Riemannian $n$-manifold (generalizing R. McCann's original work) that underlies many of the applications mentioned in talk I. We'll also examine aspects of second- and fourth- order diffusion type equations, which can be seen as gradient descents in the Riemannian geometry of probability measures.
In this third talk we'll conclude our survey of optimal transport. Applications include using Wasserstein distance to measure how distortions propogate in tomography (specifically the geodesic Radon transform on the $n$-sphere). We'll examine Max-K. von Renesse's result that reinterprets the Schr\"oedinger equation as a second order equation in the Riemannian geometry of probability measures.
The theory of optimal transport is concerned with phenomena arising when one matches two mass distributions in a most economic way, minimizing transportation cost of moving mass from one location to another. We consider an optimal transportation problem with costs satisfying certain type of degenerate curvature condition. This condition is a slightly stronger but still degenerate version of the Ma-Trudinger- Wang condition for regularity of optimal transport maps. We explain a continuity result of optimal maps with rough data on local and global domains. If time permits, we will also explain a connection to Principal- Agent problem in microeconomics. These reflect joint work in progress with Alessio Figalli and Robert McCann.
We explore possibilities and limitations of a purely topological approach to the celebrated Dvoretzky Theorem via studying its non-integrable version. The talk is based on a joint work with S. Ivanov and S. Tabachnikov.