Some Analytical Aspects of the Theory of Integrable Systems

“可积系统”这个术语通常指的是数学对象，最常见的是微分方程式，具有特殊的对称性质，可以非常详细地研究它们，有时甚至可以用封闭的形式来求解它们。这类可积系统包括几个基本的自然方程，可积系统的数学基础可以追溯到刘维尔、高斯和庞加莱的经典著作。目前，可积系统理论已经成为一个不断发展的领域，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。除了传统的微分方程域之外，可积技术在各种领域中变得普遍，如正交多项式、弦论、计数拓扑学、统计力学、随机过程、量子信息学和数论。项目中考虑的许多问题都与这些学科有直接的联系。这一研究项目延续了首席研究员在可积系统理论方面的长期研究努力。该项目的主要目标是解决从随机矩阵理论和精确可解量子模型理论的最新发展中出现的以下两个方向：(A)研究等单形tau函数、它们的渐近性、Fredholm行列式表示及其与共形场论的关系；(B)在研究随机矩阵和统计力学中的临界现象时出现的Toeplitz、Hankel和Fredholm型行列式的渐近分析。这些方向中的每一个都由一组具体的问题表示，它们将在相同的分析框架内进行研究，即Riemann-Hilbert方法。第一个方向的主要焦点是计算Painleve tau函数族的渐近联系公式的长期存在的问题，包括恒定的前因子。第二个方向的主要问题是各种类型的双重标度和过渡极限--这是随机矩阵理论和统计力学中的主要问题之一。

The term, "Integrable Systems," usually refers to mathematical objects, most often differential equations, with special symmetry properties that allow them to be studied in a very detailed way and sometimes even to solve them in a closed form. The class of integrable systems includes several fundamental equations of nature and the mathematical foundations of integrable systems go back to classical works of Liouville, Gauss, and Poincare. Currently, the theory of integrable systems has become an expanding area which plays an increasingly important role as one of the principal sources of new analytical and algebraic ideas for many branches of modern mathematics and theoretical physics. Simultaneously, it provides an efficient analytical tool for the study of some of the fundamental mathematical models arising in modern nonlinear science and technology. In addition to the traditional domain of differential equations, integrable techniques are becoming common in such diverse fields as orthogonal polynomials, string theory, enumerative topology, statistical mechanics, random processes, quantum informatics, and number theory. Many of the problems considered in the project have direct connections with these disciplines. This research project continuous the principal investigator's long term research efforts in the theory of integrable systems. The principal goal of the project is to address the following two directions which have emerged from recent developments in random matrix theory and in the theory of exactly solvable quantum models: (a) The study of the isomonodromic tau functions, their asymptotics, Fredholm determinant representations and their relations to the conformal field theory; (b) The asymptotic analysis of Toeplitz, Hankel, and Fredholm determinants arising in the study of critical phenomena in random matrices and statistical mechanics. Each of these directions is represented by a collection of concrete problems, and they will be investigated within the same analytical framework, viz., the Riemann-Hilbert method. The main focus of the first direction is on the long standing question of evaluation the asymptotic connection formulae, including the constant pre-factors, for the generic families of Painleve tau functions. The principal concern of the second direction is various types of double scaling and transitional limits - topics which are among the principal questions in both the theory of random matrices and in statistical mechanics.