Some Aspects of Random Matrices and Integrable Systems

这个建议的目的是研究随机矩阵理论和可积系统的各个方面。该项目的一部分是研究随机矩阵理论中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。这些问题的动机是统计和可积微分方程。长期的目标是要了解一个清晰的图片的普遍性性质的随机矩阵理论。随机矩阵理论描述了一系列广泛的数学对象，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。本文旨在研究随机矩阵的一些内在性质，以提高我们对这一领域的理解。

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.