Some Aspects of Random Matrices and Integrable Systems

本建议的目的是研究随机矩阵理论和可积系统的各个方面。项目的一部分是研究随机矩阵理论中出现的各种分布函数的渐近性质，利用它们与可积系统的联系。该项目的另一部分涉及普遍性问题。这些都是由统计学和可积微分方程中的问题引起的。长期目标是对随机矩阵理论的普适性有一个清晰的认识。随机矩阵理论描述了数学中广泛的对象，如数论、组合学、概率论，以及其他科学学科，如统计学、物理学、经济学、金融学和电气工程。本提案旨在研究从这些应用中激发的随机矩阵的一些内在性质，以提高我们对该领域的理解。

The aim of this proposal is to study various aspects of random matrix theory and integrable systems. One pat of the project is to study asymptotic properties of various distribution functions arising in random matrix theory using their connection to integrable systems. Another part of the project concerns on the universality questions. These are motivated by problems in statistics and integrable differential equations. The long term goal is to understand a clear picture of the universality property of random matrix theory. Random matrix theory describes a wide array of objects in mathematics such as number theory, combinatorics, probability, as well as in other disciplines of science like statistics, physics, economics, finance and electrical engineering. This proposal is aimed to study some intrinsic properties of random matrices motivated from such applications to improve our understanding of the field.