Riemann-Hilbert Problems, Integrable Systems and Random Matrix Theory

PI计划研究数学、应用数学和物理方面的各种问题。所考虑的所有问题在本质上都是渐近的，因为这些问题依赖于一个大参数，如时间或空间，或一个小参数，如摄动强度。主要问题是确定当参数分别趋近于无穷或趋近于零时系统的行为。事实证明，我们所考虑的问题有一个黎曼-希尔伯特表示法，它为这些问题提供了一个非交换的类比，经典特殊函数的积分表示法，如贝塞尔函数或艾里函数等。正如经典的特殊函数可以用最陡下降/平稳相法进行渐近分析一样，Riemann-Hilbert问题也可以用PI和X.Zhou在1993年引入的非线性最陡下降法进行分析。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面的激光模式的螺旋渐近性；随着各向异性和场强的变化，XY自旋-1/2链空性形成概率的渐近性；无限维可积系统的摄动理论，如摄动非线性薛定谔方程，在孤子存在的聚焦情况下。此外，PI将考虑随机矩阵理论和Toeplitz和Hankel行列式的渐近性问题。数学、应用数学和物理中的许多问题都可以被重新表述为黎曼-希尔伯特问题，这是一个引人注目的、出乎意料的事实。这使得分析它们的行为具有与19世纪的经典问题（如电和磁）相同的效率和准确性成为可能。特别是，各种随机矩阵系可以用黎曼-希尔伯特方法进行分析。随机矩阵本身为一系列的问题提供了模型，从重核的中子散射到临界线上的黎曼-泽塔函数的零点。例如，在交通理论方面，PI和他的合作者最近展示了墨西哥Cuernevaca的公交系统是如何用随机矩阵理论来描述的：这种公交系统具有特殊的特征，在拉丁美洲的许多地方都有使用。可以用随机矩阵理论建模的问题列表包括组合学、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黎曼-希尔伯特方法和随机矩阵理论的文本，这些文本应该可供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.