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

该奖项为CBMS会议"戴森-施温格方程，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日在纽约市的哥伦比亚大学举行。 这次会议的特色是数学家爱丽丝·吉奥内的十个系列讲座。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行为，以及围绕它们的普遍波动。矩阵是数学和所有科学中的基本对象-它们编码空间变换，物理学中的原子描述，化学反应链和化学中的概率，以及实验科学中的一系列观察，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，并且相关矩阵最终具有结构化但随机的条目。最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从噪声中分离出来？主要系列讲座将报告在解决这些类型的问题的一些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发言。 此外，我们还将安排每日辅导课和习题课，并提前提供课堂讲稿，帮助学生做好充分的准备。在过去的二十年里，随机矩阵或相关模型（如随机平铺）的高维渐近性在概率论、数学物理和统计力学中一直备受关注。由于这些模型高度相关，基于自变量的经典方法失败了。由于定律的特殊形式，如行列式定律，特殊情况已经被详细研究。这些讲座将讨论一般类的模型使用所谓的戴森-施温格方程和推广。十个讲座涵盖以下主题：大数定律在随机矩阵和浓度的措施;中心极限定理在一个切割的情况下;推广到离散设置的瓷砖模型;拓扑扩展大N渐近的分区函数;和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 Bourgeon（纽约大学）、Vadim Gorin（马萨诸塞州理工学院）和Antti Knowles（苏黎世联邦理工学院）还将发表演讲。

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