Mathematical Sciences: Some Analytical Aspects of the Theory of the Integrable Systems

DMS-9501559 PI：ITS将研究数学物理中的几个主题。其中包括量子精确可解模型关联函数的渐近分析的Riemann-Hilbert方法，Painleve方程理论中的等单调方法及其在单调反问题中的应用，以及矩阵模型和正交多项式的渐近分析的Riemann-Hilbert格式。这项研究涉及到非线性可积方程理论，它首次被用来通过所谓的逆散射方法来积分Korteweg-de Vries方程。在过去的二十年里，这种方法已经发展成为数学物理的一个重要分支，是许多现代数学分支，包括量子群论和特殊函数理论，新的解析和代数思想的主要来源之一。

DMS-9501559 PI: Its Its will investigate several topics in mathematical physics. These include Riemann-Hilbert approach to the asymptotic analysis of the correlation functions of quantum exactly solvable models, the isomonodromy method in the theory of Painleve equations and its application to the inverse monodromy problems, and Riemann-Hilbert scheme of the asymptotic analysis of the matrix models and orthogonal polynomials. The research involves the theory of nonlinear integrable equations, which was first used to integrate the Korteweg-de Vries equation via the so called inverse scattering method. During the last twenty years this method has been developed into an important branch of mathematical physics and is one of the principal sources of new analytical and algebraic ideas of many branches of modern mathematics, including quantum group theory and the theory of special functions.