The concept of mass transport was introduced by the French geometer G. Monge in 1781 and rediscovered and crucially developed and applied to areas in economics by L. Kantorovich, eventually earning him the Nobel prize in 1939. The renaissance of the classical Monge-Kantorovich mass transport topic in the late 1980's is attributed to independent developments by the French mathematician Y. Brenier (studying fluid dynamics), U.S. mathematician J. Mather in dynamical systems, and British meteorologist Mike Cullen. The mathematics of the optimal transport and its extensions has made a tremendous impact on several fields including calculus of variations, functional analysis, geometry, and probability. In recent years the geometry of optimal transport, particularly its link with the so-called Ricci curvature in Riemannian geometry, and connections to functional and isoperimetric inequalities has been extensively investigated resulting in deep and beautiful connections to the above-mentioned topics. The new book by the recent Fields medalist C. Villani is a testament to this explosive development. What is currently lacking in our understanding, and is actively being sought by various isolated groups, is the analogous development of the topic in discrete spaces -- developing appropriate "discrete calculus" on graphs and finite Markov chains. The PI and his collaborators (including students and postdocs) have identified several concrete directions to make progress, as well as find new applications, in this important and exciting topic.
In recent collaboration, with various collaborators, the PI has initiated a fruitful line of frontier research in developing discrete aspects of the exciting topic of Optimal Transport & Applications. Besides strengthening and refining classical notions, this work identifies several interesting new directions to pursue -- new notions of (Brunn-Minkowski) convexity in graphs, new concentration inequalities (dimension-free, infimum-convolution and transport-entropy inequalities) refining Talagrand's convex distance concentration (and extensions by Marton and others) on non-product spaces, as well as connections to classical sumset inequalities in additive combinatorics. Much of this is motivated by the attempts to understand appropriate metrics and geodesics in optimal transport (of measures) on discrete spaces. Developing the necessary discrete calculus and relating the recently introduced notions of discrete Ricci curvature, displacement convexity and Wasserstein-type metrics in finite graphs and Markov chains, is a technically as well as conceptually challenging objective of the proposal. Connections to other functional inequalities (such as versions Talagrand, Marton transport inequalities) and their equivalent dual formulations provide an important motivation. A second objective is to explore the full extent of applications of the methods developed and the recent theorems proved. While classical theorems (such as the log-Sobolev inequality for the Gaussian measure, Strassen's martingale existence theorem, Talagrand's theorem on the symmetric group) have already been re-derived in the recent initiative of the PI and his collaborators, the full potential needs to be more thoroughly investigated. Concentration inequalities on the noncrossing partition lattice and consequences are a concrete example of new questions that have arisen from this investigation. Another goal of the PI is to compare and constrast the various independent suggested notions of (RiccI) curvature and displacement convexity in discrete spaces.
