Riemann-Hilbert Problems, Integrable Systems and Random Matrix Theory

黎曼-希尔伯特问题、可积系统和随机矩阵理论

• 批准号: 1001886
• 负责人: Percy Deift
• 金额: $ 16.9万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001886&HistoricalAwards=false
• 关键词: Riemann; Hilbert; Problems; Integrable; Systems

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.

PI计划解决从数学、应用数学到物理的各种问题。所考虑的所有问题本质上都是渐近的，因为问题依赖于大参数，如时间或空间，或小参数，如扰动强度。主要问题是确定参数(S)为无穷大或为零时系统的行为。结果表明，所讨论的问题具有Riemann-Hilbert表示，它为这些问题提供了经典特殊函数的积分表示的非对易模拟，如Bessel函数或Ary函数等.正如经典特殊函数可以用最陡下降/稳定相法进行渐近分析一样，Riemann-Hilbert问题也可以用Pi和X.周在1993年提出的非线性最陡下降法来分析.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面的激光器模式的螺旋渐近性；随着各向异性和场强的变化，XY自旋-1/2链的空洞形成几率的渐近性；当孤子存在时，无限维可积系统的微扰理论，如扰动的非线性薛定谔方程。此外，PI将考虑随机矩阵理论以及Toeplitz和Hankel行列式的渐近问题。数学、应用数学和物理学中的许多问题都可以改称为黎曼-希尔伯特问题，这是一个不同寻常且出人意料的事实。这使得能够像19世纪的电学和磁学等经典问题一样，以同样的效率和准确性来分析它们的行为。特别地，各种随机矩阵系综可以用Riemann-Hilbert方法来分析。随机矩阵本身提供了一系列特殊问题的模型，从中子在重核上的散射，到临界线上Riemann-Zeta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如何用随机矩阵理论来描述墨西哥库尔内瓦卡的公共汽车系统：这种公共汽车系统具有特殊的特征，并在拉丁美洲的许多地区使用。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等。PI和他的合作者还参与了关于Riemann-Hilbert方法和随机矩阵理论的各种文本的撰写，这些文本应该对整个科学光谱的研究人员都是可用的。