Some Analytical Aspects of the Theory of Integrable Systems

这项研究主要集中在与Riemann-Hilbert方法和等序法相关的可积系统理论的分析方面，所考虑的问题包括精确可解量子场和统计力学模型相关函数的渐近分析，正交多项式理论中双尺度极限的研究，描述随机矩阵理论和随机置换的分布函数的可积系统的分析，以及Painleve‘和KdV型可积微分方程解的全局渐近分析。可积系统理论是一个不断扩展的领域，它作为现代数学和理论物理许多分支新的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，它为研究现代非线性科学和技术中出现的一些基本数学模型提供了一种有效的分析工具。具体地说，上述研究方向涉及构成以下领域的理论基础的数学模型：凝聚态、高能和等离子体物理、非线性流体力学和高比特率电信系统。在这个项目中，特别注意分析随机矩阵理论的分布，它支配着不服从通常的经典概率定律的大系统的统计性质。这类系统出现在应用科学和技术的许多不同领域，包括重核、聚合物生长、高维数据分析和某些渗流过程。

The research project is focused on the analytical aspects of the theory ofintegrable systems related to the Riemann-Hilbert and isomonodromy methods.The problems under consideration include the asymptotic analysis of thecorrelation functions of exactly solvable quantum field and statisticalmechanics models, the investigation of the double scaling limits in the theory of orthogonal polynomials, the analysis of the integrable systems describing the distribution functions of random matrix theory and random permutations, and the global 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 mathematics andtheoretical physics. Simultaneously, it provides an efficient analytic toolfor the study of some of the fundamental mathematical models arising in modern nonlinear science and technology. Specifically, the research directions indicated above deal with the mathematical models which form the theoretical basis for the following fields: condensed matter, high energy and plasma physics, nonlinear hydrodynamics, and high-bit rate telecommunication systems. A special attention in the project is given to the analysis of the distributions of random matrix theory which govern statistical properties of the large systems which do not obey the usual laws of classical probability. This kind of systems appears in many different areasof applied science and technology including heavy nuclei, polymer growth, high-dimensional data analysis, and certain percolation processes.