Riemann-Hilbert Problems, Integrable Systems and Random Matrix Theory

国际数学联合会将继续分析数学和数学物理中的各种渐近问题。Riemann-Hilbert问题的最速下降法将继续发挥关键作用。特别是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性问题**多重正交多项式的渐近性，以期证明某些区分常数**Toeplitz和Hankel行列式的渐近评估**在分析玻色-爱因斯坦凝聚**时出现的那种统计实验，证明了随机矩阵理论在数学中的普遍性。在过去的五十年里，最引人入胜的科学发展之一是发现了非常广泛的数学和物理现象是由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核上的散射，以及复平面中临界线上Riemann Zeta函数的零点的统计。最近，人们发现随机矩阵理论也描述了现实世界中的问题，例如墨西哥库尔内瓦卡市的公交车调度问题，曼哈顿电话簿中电话号码的分配，以及一种名为耐心排序的纸牌的统计。PI工作的一个中心重点将涉及分析随机矩阵理论本身，以及将随机矩阵理论应用于数学、物理和“现实世界”中的新问题。

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".