Some Analytical Aspects of the Theory of Integrable Systems

术语“可积系统”通常指的是数学对象，最常见的是微分方程，具有特殊的对称性，使它们能够以非常详细的方式进行研究，有时甚至以封闭的形式求解。 可积系统的类包括几个基本方程的性质和数学基础的可积系统可以追溯到经典著作的刘维，高斯和庞加莱。 目前，可积系统理论已成为一个不断扩大的领域，作为现代数学和理论物理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领域，可积技术在正交多项式、弦理论、枚举拓扑、统计力学、随机过程、量子信息学和数论等不同领域变得越来越普遍。 项目中考虑的许多问题都与这些学科有直接联系。本研究计画延续了本研究者在可积系统理论上的长期研究成果。该项目的主要目标是解决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）研究等单道τ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）在随机矩阵和统计力学的临界现象研究中产生的Toeplitz、Hankel和Fredholm行列式的渐近分析。 这些方向中的每一个都由一系列具体问题代表，它们将在同一分析框架内进行研究，即，黎曼-希尔伯特方法 第一个方向的主要重点是长期存在的问题，评估的渐近连接公式，包括常数前因子，为通用家庭的Painleve tau函数。主要关注的第二个方向是各种类型的双重标度和过渡限制-主题是其中的主要问题在理论的随机矩阵和统计力学。

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.