CBMS Conference: Dyson-Schwinger Equations, Topological Expansions, and Random Matrices

CBMS 会议：Dyson-Schwinger 方程、拓扑展开式和随机矩阵

• 批准号: 1642595
• 负责人: Ivan Corwin
• 金额: $ 3.78万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1642595&HistoricalAwards=false
• 关键词: CBMS; Conference; Dyson; Schwinger; Equations

This award provides support for the CBMS conference "Dyson-Schwinger Equations, Topological Expansions, and Random Matrices," which will run from August 28 to September 1, 2017 at Columbia University in New York City. The conference features a series of ten lectures by mathematician Alice Guionnet. Probability, as a field, seeks to understand how large, complex random systems manifest deterministic behaviors, and universal fluctuations around them. Matrices are fundamental objects in mathematics and all of science -- they encode transformations of space, descriptions of atoms in physics, chains of chemical reactions and probabilities in chemistry, and series of observations in experimental sciences, and they are vital in signal processing. In many applications, there is noise present in the system under study, and the associated matrices end up having structured but random entries. The fundamental challenge becomes to understand the effect of noise on these structures. How does it change the key properties of these objects, and how can statistical methods be developed to separate the signal from the noise? The principal lecture series will report on some of the most exciting new advances in tackling these types of questions. The meeting also will include presentations by other experts in the subject area. There will also be daily tutorials and problem sessions, with lecture notes provided in advance to help students prepare adequately.Understanding the large-dimension asymptotics of random matrices or related models such as random tilings has been of great interest for the last twenty years within probability, mathematical physics, and statistical mechanics. Because such models are highly correlated, classical methods based on independent variables fail. Special cases have been studied in detail thanks to specific forms of the laws, such as determinantal laws. These lectures will discuss a general class of models using the so-called Dyson-Schwinger equations and generalizations. The ten lectures cover the following subjects: law of large numbers in random matrices and concentration of measure; central limit theorem in the one-cut case; generalization to the discrete setting of tiling models; topological expansions for large-N asymptotics of partition functions; and generalization to several-cut models. Additional talks will be presented by Charles Bordenave (University of Toulouse), Gaetan Borot (Max Planck Institute for Mathematics, Bonn), Paul Bourgade (New York University), Vadim Gorin (Massachusetts Institute of Technology), and Antti Knowles (ETH Zurich).The website for this conference is: http://www.math.columbia.edu/department/probability/seminar/Guionnet.html

该奖项支持将于2017年8月28日至9月1日在纽约哥伦比亚大学举行的CBMS会议《Dyson-Schwinger方程、拓扑展开和随机矩阵》。这次会议以数学家爱丽丝·古尼特的十个系列讲座为特色。概率作为一个领域，试图理解大型、复杂的随机系统如何表现出确定性的行为，以及它们周围的普遍波动。矩阵是数学和所有科学中的基本对象--它们编码空间的变换，物理中的原子描述，化学反应的链和化学中的概率，以及实验科学中的一系列观测，它们在信号处理中至关重要。在许多应用中，在所研究的系统中存在噪声，并且相关矩阵最终具有结构化但随机的条目。根本的挑战是了解噪声对这些结构的影响。它如何改变这些物体的关键属性，以及如何开发统计方法来将信号与噪声分开？主要的系列讲座将报道一些在解决这类问题方面最令人兴奋的新进展。会议还将包括该主题领域其他专家的介绍。课程还包括每天的教程和问题课，提前提供课堂讲稿，帮助学生充分准备。了解随机矩阵或相关模型的高维渐近性，如随机分块，在过去的二十年里一直是概率论、数学物理和统计力学领域的研究热点。由于这些模型高度相关，基于自变量的经典方法失败了。由于法律的具体形式，如行列式法律，对特殊情况进行了详细的研究。这些讲座将讨论使用所谓的Dyson-Schwinger方程和推广的模型的一般课程。十个讲座涵盖以下主题：随机矩阵中的大数定律和测度集中；单割情况下的中心极限定理；对平铺模型离散设置的推广；配分函数大N渐近的拓扑展开式；以及对多割模型的推广。查尔斯·博尔德纳夫(图卢兹大学)、盖坦·博罗(波恩马克斯·普朗克数学研究所)、保罗·伯加德(纽约大学)、瓦迪姆·戈林(麻省理工学院)和安蒂·诺尔斯(苏黎世理工学院)将作其他演讲。这次会议的网站是：http://www.math.columbia.edu/department/probability/seminar/Guionnet.html