Title: Optimal Mass Transport and Curvature Bounds
Abstract: Have you ever been in a hurry at the grocery store? Have you ever wondered if you are collecting the items on your list in the most efficient way possible? This is a discrete case of the fundamental problem of optimal mass transport theory. As a consequence of its many and varied applications in both pure and applied mathematics, the problem of optimal mass transport theory has received considerable attention during the past few years. (Indeed, Cédric Villani, one of the prominent researchers in this area, was awarded the Fields Medal in 2010.) On the pure side, the theory of optimal mass transport has found important connections to nonlinear partial differential equations (such as the critically-important Monge-Ampère equations that appear within its foundations), dynamical systems, inequalities (including isoperimetric inequalities), and geometry. On the applied side, partial answers to the transport question have had significant implications in such fields as economics, statistics, computer vision, physics, fluid dynamics, engineering, biology, and even meteorology.
In this talk, we will introduce the basic notions of optimal mass transport theory as well as discuss some interactions between optimal mass transport theory and the geometry of the underlying (and other related) spaces. More particularly, we will present some results concerning the inherited geometric properties of the 2-Wasserstein metric space of probability measures (over its parent space) and state a special case of the so-called curvature-dimension condition arising in the study of optimal transport over Riemannian manifolds. We will also explore the possibility of extending L. Caffarelli’s celebrated contraction theorem to the setting of the standard sphere as well as examine a curvature bound obtained very recently by A. Kolesnikov. We will end with a generalization of a part of E. Calabi’s work concerning Hessian metrics.
The majority of this presentation should be accessible to all graduate students, especially those with coursework in differential geometry. All are welcome to attend.