Some Analytical Aspects of the Theory of Integrable Systems

该研究项目的重点是与Riemann-Hilbert和isomonodromy方法相关的可积系统理论的分析方面。所考虑的问题包括：（a）随机矩阵和正交多项式理论中的双标度极限及其在二维量子引力和计数拓扑中的应用的研究：（B）与精确可解量子场和统计力学模型的关联函数有关的Toeplitz和Fredholm行列式和重积分的渐近分析;（c）Painleve和KdV型可积微分方程解的整体渐近分析的有关问题。可积系统理论是一个不断扩展的领域，它作为现代物理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，发挥着越来越重要的作用。同时，它也为研究现代非线性科学技术中出现的一些基本物理模型提供了一种有效的分析工具。具体地说，上述研究方向涉及形成以下领域理论基础的数学模型：凝聚态物理、高能物理和等离子体物理、非线性流体力学、高比特率电信系统和信息科学。在这个项目中，特别注意随机矩阵理论的分布的分析，随机矩阵理论支配着不遵守经典概率通常规律的大系统的概率性质。这类系统出现在许多不同的应用科学和技术领域，包括重核、聚合物生长、高维数据分析和某些纯化过程。同时，随机矩阵作为一种极其强大的分析工具，在弦理论、计数拓扑学和数论的某些基本方面等纯理论领域中发挥着越来越重要的作用。

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.