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

该奖项为CBMS会议“Dyson-Schwinger方程，拓扑展开和随机矩阵”提供支持，该会议将于2017年8月28日至9月1日在纽约市哥伦比亚大学举行。这次会议的特色是由数学家爱丽丝·吉奥内（Alice Guionnet）主持的一系列十场讲座。概率论，作为一个领域，试图理解大的，复杂的随机系统如何表现出确定性行为，以及它们周围的普遍波动。矩阵是数学和所有科学的基本对象——它们编码空间的变换，物理学中的原子描述，化学中的化学反应链和概率，以及实验科学中的一系列观察结果，它们在信号处理中至关重要。在许多应用中，所研究的系统中存在噪声，而相关的矩阵最终具有结构化但随机的条目。最根本的挑战是理解噪音对这些结构的影响。它是如何改变这些物体的关键属性，以及如何开发统计方法来分离信号和噪声？主要系列讲座将报告在解决这类问题方面一些最令人兴奋的新进展。会议还将由该主题领域的其他专家作报告。每天也会有辅导课和习题课，并提前提供课堂讲稿，帮助学生做好充分的准备。了解随机矩阵或相关模型（如随机拼接）的大维渐近性在过去二十年中在概率、数学物理和统计力学领域引起了极大的兴趣。由于这些模型高度相关，基于自变量的经典方法失败了。由于这些定律的特定形式，如决定性定律，特殊情况得到了详细的研究。这些讲座将讨论使用所谓的戴森-施温格方程和推广的一类一般模型。十堂课的内容包括：随机矩阵的大数定律和测度的集中；单切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；并推广到几个切割模型。Charles Bordenave（图卢兹大学），Gaetan Borot（波恩马克斯普朗克数学研究所），Paul Bourgade（纽约大学），Vadim Gorin（麻省理工学院）和Antti Knowles（苏黎世联邦理工学院）将进行其他演讲。会议的网址是：http://www.math.columbia.edu/department/probability/seminar/Guionnet.html

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