Displacement Convexity, Curvature and Concentration in Discrete Settings

大众运输的概念是由法国几何学家G.蒙格于1781年提出的，并由L.坎托洛维奇重新发现并关键地发展和应用于经济学领域，最终为他赢得了1939年的诺贝尔奖。20世纪80年代末，经典的蒙格-康托洛维奇质量输运理论的复兴归功于法国数学家Y·布雷尼尔(研究流体力学)、美国动力系统数学家J·马瑟和英国气象学家迈克·卡伦的独立发展。最优运输的数学及其推广对变分、泛函分析、几何和概率等领域产生了巨大的影响。近年来，最优传输的几何，特别是它与黎曼几何中所谓的Ricci曲率的联系，以及与泛函和等周不等式的联系被广泛地研究，从而导致了对上述主题的深入而美丽的联系。最近的菲尔兹奖牌获得者C·维拉尼的这本新书证明了这一爆炸性的发展。我们目前所缺乏的理解，并正在被各种孤立的群体积极寻求的，是在离散空间中类似的主题的发展--在图和有限马尔可夫链上发展适当的“离散演算”。PI和他的合作者(包括学生和博士后)已经确定了在这一重要而令人兴奋的主题中取得进展并找到新应用的几个具体方向。在最近的合作中，PI与各种合作者合作，在开发令人兴奋的最优运输和应用这一令人兴奋的主题的离散方面开创了一系列卓有成效的前沿研究。除了加强和提炼经典概念外，这项工作还确定了几个有趣的新方向--图中(Brunn-Minkowski)凸性的新概念，新的集中不等式(无量纲、下确界卷积和传输熵不等式)，提炼了TALAGRAND在非乘积空间上的凸距离集中(以及Marton和其他人的扩展)，以及与加法组合学中经典求和集不等式的联系。这在很大程度上是因为试图理解离散空间上(测量的)最佳运输中的适当度量和测地线。发展必要的离散演算，并将最近引入的有限图和马尔可夫链中的离散Ricci曲率、位移凸性和Wasserstein型度量概念联系起来，是该提议的一个技术和概念上的挑战。与其他泛函不等式(如版本TALAGRAND，Marton传输不等式)的联系及其等价的对偶公式提供了一个重要的动机。第二个目标是探索所开发的方法和最近证明的定理的全部应用范围。虽然经典的定理(如关于高斯测度的LOG-Soblev不等式，Strassen的鞅存在定理，关于对称群的TALAGRAND定理)已经在PI和他的合作者最近的倡议中被重新推导，但全部潜力还需要更深入的研究。不相交划分格上的集中不等式及其结果是从这项研究中出现的新问题的一个具体例子。PI的另一个目标是比较和对比离散空间中各种独立的(Ricci)曲率和位移凸性概念。

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.