Displacement Convexity, Curvature and Concentration in Discrete Settings

质量传递的概念是由法国几何学家G. Monge在1781年提出的，被L. Kantorovich重新发现并关键地发展并应用于经济学领域，最终使他在1939年获得诺贝尔奖。经典的Monge-Kantorovich质量输运问题在20世纪80年代后期的复兴归功于法国数学家Y. Brenier（研究流体动力学）、美国动力学系统数学家J. Mather和英国气象学家Mike Cullen的独立发展。最优输运的数学及其扩展在几个领域产生了巨大的影响，包括变分学、泛函分析、几何和概率论。近年来，最优输运的几何，特别是它与黎曼几何中所谓的里奇曲率的联系，以及与泛函和等周不等式的联系得到了广泛的研究，导致了与上述主题的深刻而美丽的联系。最近获得菲尔兹奖的C.维拉尼（C. Villani）的新书证明了这种爆炸性的发展。目前在我们的理解中缺乏的是，各种孤立的团体正在积极寻求在离散空间中类似的主题发展——在图和有限马尔可夫链上发展适当的“离散微积分”。PI和他的合作者（包括学生和博士后）已经确定了几个具体的方向来取得进展，并在这个重要而令人兴奋的主题中找到新的应用。在最近的合作中，与不同的合作者，PI在开发最佳运输和应用这一激动人心的主题的离散方面发起了卓有成效的前沿研究。除了加强和改进经典概念，这项工作还确定了几个有趣的新方向——图中的（布伦-闵可夫斯基）凸性的新概念，新的集中不等式（无维，无穷大卷积和输运熵不等式），改进了Talagrand在非积空间上的凸距离集中（以及Marton和其他人的扩展），以及与加性组合中的经典sumset不等式的联系。这在很大程度上是为了理解离散空间中最佳运输（度量）的适当度量和测地线。发展必要的离散微积分，并将最近引入的离散里奇曲率、位移凸性和有限图和马尔可夫链中的wasserstein型度量的概念联系起来，是该提案在技术上和概念上具有挑战性的目标。与其他函数不等式（如Talagrand、Marton输运不等式）及其等效对偶公式的联系提供了一个重要的动机。第二个目标是探索开发的方法和最近证明的定理的全部应用范围。虽然经典定理（如高斯测度的log-Sobolev不等式，Strassen的鞅存在定理，关于对称群的talagand定理）已经在PI和他的合作者最近的倡议中被重新推导出来，但其全部潜力需要更彻底的研究。非交叉分区格上的浓度不平等及其结果是本研究产生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（RiccI）曲率和位移凸性的各种独立的建议概念。

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.