Riemann-Hilbert Problems, Integrable Systems and Random Matrix Theory

摘要deift PI将继续他在数学和数学物理中各种渐近问题的分析方面的工作。黎曼-希尔伯特问题的最陡下降法将继续发挥关键作用。特别地，PI将考虑**支持孤子解的特定无限维可积系统的摄动理论**随机矩阵理论中的普适问题**多个正交多项式的渐近性，以证明某些区分常数的无理性** Toeplitz和Hankel行列式的渐近评价。在分析玻色-爱因斯坦凝聚的统计实验中出现的那种，这些实验证明了随机矩阵理论在数学、物理和“现实世界”中的普遍存在。在过去的五十年里，最令人着迷的科学发展之一是发现了一个非常广泛的数学和物理现象是由一个随机矩阵的特征值来建模的。特别是，随机矩阵理论描述了中子在大原子核上的散射，以及黎曼ζ函数在复平面上临界线上的零点统计。最近，人们发现随机矩阵理论也可以描述“现实世界”中的问题，比如墨西哥库尔内瓦卡市的公交调度问题，曼哈顿电话簿中的电话号码分布，以及一种被称为耐心排序的纸牌游戏的统计。PI工作的中心焦点将涉及随机矩阵理论本身的分析，以及随机矩阵理论在数学、物理和“现实世界”中的新问题的应用。

AbstractDeiftThe PI will continue his work on the analysis of various asymptoticproblems in mathematics and mathematical physics. The steepest-descent method for Riemann-Hilbert problems will continue to play a key role.In particular, the PI will consider** the perturbation theory of specific infinite-dimensional integrable systems which support soliton solutions** universality problems in random matrix theory** asymptotics for multiple orthogonal polynomials, with a view to provingthe irrationality of certain distinguished constants** the asymptotic evaluation of Toeplitz and Hankel determinants, of the kind that arise in the analysis of Bose-Einstein condensation** statistical experiments which demonstrate the prevalence of random matrix theory in mathematics, physics and also in the "real world".One of the most fascinating scientific developments over the last fifty years has been the discovery that an extraordinarily wide variety of mathematical and physical phenomena are modelled by the eigenvalues of a random matrix. In particular, random matrix theory describes the scattering of neutrons off large nuclei, as well as the statistics of the zeros of the Riemann zeta function on the critical line in the complex plane. More recently, it has been discovered that random matrix theory also describes problems in the "real world", such as the bus scheduling problem in the city of Cuernevaca in Mexico, the distribution of telephone numbers in the Manhattan telephone directory, as well as the statistics of a form of solitaire called patience sorting. A central focus of the PI's work will involve the analysis of random matrix theory per se, as well as the application of random matrix theory to new problems in mathematics, physics and the "real world".