Riemann-Hilbert Problems, Integrable Systems and Random Matrix Theory

PI计划在可积系统的一般领域研究各种问题，特别是渐近问题。两个关键工具将在PI的工作中发挥重要作用：一方面是Riemann-Hilbert问题（RHP）和非线性、非交换的最速下降法，另一方面是随机矩阵理论（RMT）。RHP可以被看作是经典特殊函数的标量积分表示的自然的、非交换的推广，并且正如经典的最速下降法允许人们评估经典函数的渐近行为一样，非线性的、非交换的最速下降法也允许人们评估现代可积系统的渐近行为，例如KdV方程和Painleve方程，还有很多人RMT，特别是随机矩阵的特征值，提供了一个非常通用的工具来描述各种各样的随机现象的行为时，潜在的变量不再是独立的。PI将用这些工具解决的问题包括：当菲涅耳数趋于无穷大时激光模式的渐近行为;当n，N趋于无穷大时大小为N的随机n带矩阵的行为;一般正交和辛矩阵系综的普适性;聚焦非线性微扰理论线性薛定谔方程在线。随机矩阵的特征值为物理学中的各种统计现象提供了定量模型，这是一个非凡的事实，纯数学和应用数学以及工程学。RMT的应用范围从核散射理论中的共振分析，到解析数论中临界线上黎曼zeta函数零点的分析。非常奇怪，令人惊讶的是，甚至在墨西哥城市库埃尔纳瓦卡的公共汽车之间的到达时间，已经被证明服从随机矩阵理论统计！RMT作为一门学科有两个组成部分：随机矩阵理论本身的发展，以及该理论在新的物理和数学问题中的应用。PI的工作将影响这两个组件。在可积系统理论中，PI特别提出的两个问题直接来自技术，即，激光模式渐近性，以及扰动对光纤传输系统的影响。PI和他的合作者也在编写针对研究人员以及进入该领域的高级研究生的RHP和RMT基础文本。

The PI plans to work on a variety of problems, particularly asymptotic problems, in the general area of integrable systems. Two key tools will play a prominent role in the PI's work: Riemann-Hilbert Problems (RHP) and the non-linear, non-commutative steepest descent method, on the one hand, and Random Matrix Theory (RMT) on the other. RHPs can be viewed as the natural, non-commutative generalization of the scalar integral representations of the classical special functions, and just as the classical steepest descent method allows one to evaluate the asymptotic behavior of the classical functions, so too the non-linear, non-commutative steepest descent method allows one to evaluate the asymptotic behavior of modern integrable systems such as the KdV equation and the Painleve equations, amongst many others. RMT, and in particular, the eigenvalues of random matrices, provide an extremely versatile tool to describe the behavior of a wide variety of stochastic phenomena when the underlying variables are no longer independent. Amongst the problems that the PI will address with these tools are: the asymptotic behavior of the modes of a laser as the Fresnel number goes to infinity; the behavior of random n-band matrices of size N as n,N go to infinity; universality for general orthogonal and symplectic matrix ensembles; the perturbation theory of the focusing Non-Linear Schroedinger Equation on the line.It is an extraordinary fact that the eigenvalues of random matrices provide a quantitative model for a wide variety of statistical phenomena from physics, pure and applied mathematics, and engineering. Applications of RMT range from the analysis of resonances in nuclear scattering theory, to the analysis of the zeroes of the Riemann zeta function on the critical line, in analytic number theory.Quite spectacularly, and surprisingly, even the arrival times between buses in the Mexican city of Cuernavaca, have been shown to obey random matrix theory statistics! RMT as a discipline has two components: the development of random matrix theory per se, and the application of the theory to new physical and mathematical problems. The work of the PI will impact both components. In the theory of integrable systems, two of the problems in particular proposed by the PI, arise directly from technology, viz., laser mode asymptotics, and the effect of perturbations on fiber optical transmission systems. The PI and his collaborators are also writing basic texts on RHPs and RMT aimed at the researchers as well as advanced graduate students entering the field.