Displacement Convexity, Curvature and Concentration in Discrete Settings

离散设置中的位移凸度、曲率和浓度

• 批准号: 1407657
• 负责人: Prasad Tetali
• 金额: $ 28.8万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1407657&HistoricalAwards=false
• 关键词: Displacement; Convexity; Curvature; Concentration; Discrete

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.

质量输运的概念是由法国几何学家G.蒙日在1781年被重新发现，并在经济学领域得到了关键性的发展和应用。康托洛维奇，最终赢得了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（研究流体动力学），美国数学家J.马瑟在动力系统，和英国气象学家迈克卡伦。 最优运输的数学及其扩展对变分法、泛函分析、几何和概率等领域产生了巨大的影响。近年来，最优运输的几何，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，已经被广泛研究，从而与上述主题产生了深刻而美丽的联系。菲尔兹奖获得者C.维拉尼是这种爆炸性发展的证明。 目前我们所缺乏的理解，并正在积极寻求各种孤立的团体，是类似的发展的主题在离散空间-发展适当的“离散微积分”的图形和有限的马尔可夫链。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，以取得进展，以及寻找新的应用，在这个重要和令人兴奋的主题。在最近的合作中，PI与各种合作者，在开发令人兴奋的主题最佳运输应用的离散方面启动了一系列富有成效的前沿研究。除了加强和完善经典的概念，这项工作确定了几个有趣的新方向追求-新的概念，图的（Brunn-Minkowski）凸性，新的集中不等式（无量纲，下卷积和输运熵不等式）细化Talagrand的凸距离浓度（和扩展的马顿和其他人）的非产品空间，以及连接到经典的和集不等式的加法组合。这在很大程度上是出于试图了解适当的度量和测地线在离散空间上的最佳运输（措施）。发展必要的离散微积分，并将最近引入的离散Ricci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，这是该提案在技术上和概念上都具有挑战性的目标。连接到其他功能的不平等（如版本Talagrand，马顿运输不平等）和他们的等价的双重制定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面应用范围。虽然经典定理（如高斯测度的对数索伯列夫不等式，斯特拉森的鞅存在定理，Talagrand的对称群定理）已经在PI及其合作者最近的倡议中重新推导，但完整的潜力需要更彻底的研究。不相交划分格上的浓度不等式及其结果是本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验证离散空间中各种独立的（RiccI）曲率和位移凸性概念。