Riemann-Hilbert Problems, Integrable Systems and Random Matrix Theory

PI计划研究数学、应用数学和物理学中的各种问题。所有的问题都是渐近性质的问题取决于一个大的参数，如时间或空间，或一个小的参数，如扰动强度。主要问题是确定系统的行为时，参数（S）分别走向无穷大，或零。结果表明，所考虑的问题有一个Riemann-Hilbert表示，它为这些问题提供了一个经典特殊函数（如Bessel函数或Airy函数等）的积分表示的非交换模拟。正如经典特殊函数可以用最速下降/驻相方法渐近分析一样，同样，Riemann-Hilbert问题也可以用PI和X.Zhou在1993年提出的非线性最速下降法来分析。PI和他的合作者要考虑的问题包括：当菲涅耳数趋于无穷大时，具有矩形平面平行反射表面的激光模式的螺旋渐近性;当各向异性和场强变化时，XY自旋1/2链的空形成概率的渐近性;微扰理论的无穷维可积系统，如微扰非线性薛定谔方程，在聚焦的情况下，孤子存在。此外，PI将考虑随机矩阵理论以及Toeplitz和Hankel行列式的渐近性问题。这是一个值得注意的，意想不到的，事实上，在数学，应用数学和物理学的各种各样的问题可以被改写为黎曼-希尔伯特问题。这使得分析它们的行为成为可能，其效率和精度与19世纪的经典问题（如电学和磁学）相同。特别地，各种随机矩阵集合可以通过Riemann-Hilbert方法来分析。随机矩阵本身为一系列非常广泛的问题提供了模型，从重核的中子散射到临界线上黎曼-泽塔函数的零点。 例如，在交通理论中，PI和他的合作者最近展示了如何用随机矩阵理论描述墨西哥Cuernevaca的公共汽车系统：这个公共汽车系统具有特殊的功能，并在拉丁美洲的许多地方使用。可以用随机矩阵理论建模的问题列表包括组合数学、多元统计、数值分析中的条件数、平铺问题、相互作用粒子系统、量子传输问题和无线通信等。 PI和他的合作者还参与编写各种关于Riemann-Hilbert方法和随机矩阵理论的文本，这些文本应该可以供整个科学领域的研究人员使用。

The PI plans to work on a variety of problems from mathematics, applied mathematics and physics. All the problems under consideration are asymptotic in nature in the sense that the problems depend on a large parameter, such as time or space, or a small parameter, such as perturbation strength. The main issue is to determine the behavior of the systems when the parameter(s) go to infinity, or to zero, respectively. It turns out thatthe problems under consideration have a Riemann-Hilbert representation which provides a non-commutative analog, for these problems, of the integral representations of the classical special functions, such as the Bessel functions or the Airy function, etc. And just as the classical special functions can be analyzed asymptotically by the steepest-descent/stationary phase method, so too the Riemann-Hilbert problems can be analyzed by the non-linear steepest-descent method introduced by the PI and X.Zhou in 1993. Amongst the problems to be considered by the PI and his collaborators are: spiral asymptotics for the modes of lasers with rectangular plane-parallel reflecting surfaces, as the Fresnel number goes to infinity; asymptotics for the Emptiness Formation Probability of the XY spin-1/2 chain, as the anisotropy and field strength vary; perturbation theory of infinite dimensional integrable systems such as the perturbed Nonlinear Schroedinger Equation, in the focusing case when solitons are present. In addition the PI will consider problems in random matrix theory and in the asymptotics of Toeplitz and Hankel determinants. It is a remarkable, and unanticipated, fact that a great variety of problems in mathematics, applied mathematics and physics can be rephrased as Riemann-Hilbert problems. This makes it possible to analyze their behavior with the same efficiency and accuracy as the classical problems, such as electricity and magnetism, of the 19th century. In particular, various random matrix ensembles can be analyzed by Riemann-Hilbert methods. Random matrices in themselves provide models for an extraordinary range of problems, from the scattering of neutrons off heavy nuclei, to the zeros of the Riemann-zeta function on the critical line. In transportation theory, for example, the PI and his collaborators recently showed how the bus system in Cuernevaca, Mexico, could be described by random matrix theory: this bus system has special features and is used in many parts of Latin America. The list of problems that can be modeled by random matrix theory includes combinatorics, multivariate statistics, condition numbers in numerical analysis, tiling problems, interacting particle systems , quantum transport problems and wireless communication, amongst many others. The PI and his collaborators are also involved in writing various texts on Riemann-Hilbert methods and also on random matrix theory that should be accessible to researchers across the scientific spectrum.