Some Analytical Aspects of the Theory of Integrable Systems

可积系统理论的一些分析方面

• 批准号: 0701768
• 负责人: Alexander Its
• 金额: $ 25.23万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701768&HistoricalAwards=false
• 关键词: Some; Analytical; Aspects; Theory; Integrable

The research project is focused on the analytical aspects of the theoryof integrable systems related to the Riemann-Hilbert and isomonodromymethods. The problems under consideration include: (a) the investigationof the double scaling limits in the theory of random matrices andorthogonal polynomials and their applications to the two-dimensionalquantum gravity and enumerative topology; (b) the asymptotic analysisof the Toeplitz and Fredholm determinants and the multiple integralsassociated with the correlation functions of exactly solvable quantumfield and statistical mechanics models; (c) the related aspects of theglobal asymptotic analysis of the solutions of integrable differential equations of the Painleve' and KdV types. The theory of integrable systems is an expanding area which plays anincreasingly important role as one of the principal sources of newanalytical and algebraic ideas for many branches of modern mathematicsand theoretical physics. At the same time, it provides an efficientanalytic tool for the study of some of the fundamental mathematicalmodels arising in modern nonlinear science and technology. Specifically, the research directions indicated above deal with themathematical models which form the theoretical basis for the followingfields: condensed matter, high energy and plasma physics, nonlinearhydrodynamics, high-bit rate telecommunication systems, and informationscience. A special attention in the project is given to the analysisof the distributions of random matrix theory which govern statisticalproperties of the large systems which do not obey the usual laws ofclassical probability. This kind of systems appear in many differentareas of applied science and technology including heavy nuclei, polymergrowth, high-dimensional data analysis, and certain percolationprocesses. Simultaneously, the random matrices are playing increasinglyimportant role, as an extremely powerful analytic apparatus, in severalpure theoretical domains such as the string theory, enumerative topologyand certain fundamental aspects of number theory.

该研究项目的重点是与Riemann-Hilbert和Isomonodromythods相关的可集成系统理论的分析方面。所考虑的问题包括：（a）随机矩阵理论和多项式多项式中的双缩放限制的研究及其在二维Quantum重力和枚举拓扑中的应用； （b）Toeplitz和Fredholm决定因素的渐近分析，以及与确切可溶剂可溶剂量子的相关函数和统计力学模型的相关函数相关的多个积分； （c）全球渐近分析的相关方面对painleve和KDV类型的可集成微分方程的解。整合系统的理论是一个扩展的领域，它是现代数学和理论物理学许多分支的新分析和代数思想的主要来源之一。同时，它为研究现代非线性科学技术产生的一些基本数学模型提供了一种有效的工具。具体而言，上面指示的研究方向涉及构成以下场地的理论基础的含义模型：凝结物质，高能量和血浆物理学，非线性水合动力学，高位率电信系统和Informationscience。该项目的特殊关注是在分析随机矩阵理论的分布的分析中，该理论控制了大型系统的统计特征，这些系统不遵守通常的典型概率定律。这种系统出现在许多应用科学和技术的不同不同之处，包括重核，聚合物生长，高维数据分析和某些PercolationProcesses。同时，作为一种极其强大的分析仪，随机矩阵在诸如字符串理论，列举拓扑和数字理论的某些基本方面等多个理论领域中起着越来越重要的作用。