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

DMS-9501559 PI: Its Its将研究数学物理中的几个主题。其中包括Riemann-Hilbert方法对量子精确可解模型相关函数的渐近分析，Painleve方程理论中的等同构方法及其在逆单问题中的应用，Riemann-Hilbert格式对矩阵模型和正交多项式的渐近分析。该研究涉及非线性可积方程理论，该理论首先用于通过所谓的逆散射法对Korteweg-de Vries方程进行积分。在过去的二十年中，这种方法已经发展成为数学物理的一个重要分支，并且是现代数学许多分支（包括量子群论和特殊函数理论）的新分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.