Some Analytical Aspects of the Theory of Integrable Systems

我们所研究的这类方程，称为可积系统，包括自然界的几个基本方程。这是一个古老而可敬的研究领域，可以追溯到刘维尔、高斯和庞加莱的经典著作。目前，可积系统理论已成为一个不断扩大的领域，它作为现代数学和理论物理的许多分支的新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外，可积技术在弦理论、统计力学、随机过程、量子信息学和数论等不同领域也变得越来越普遍。项目中考虑的许多问题与这些学科有直接的联系。事实上，该项目中具体的分析问题主要是由量子纠缠分析理论的需要所激发的，量子纠缠是一种有望在量子计算的实际实现中发挥关键作用的现象，也是量子和统计力学中对相变基本问题的分析描述所面临的挑战。本研究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中出现的各种新的可积系统理论分析问题。在本项目中，PI建议重点研究三个方向：Toeplitz和Hankel行列式的渐近分析及其在随机矩阵和统计力学中的应用，随机矩阵理论中Fredholm行列式的研究，以及非自由费米子量子自旋模型相关函数的渐近性。这些方向中的每一个都由一组具体问题来表示，它们将在相同的分析框架内进行研究，即黎曼-希尔伯特方法。

The class of equations under study, called integrable systems, includes several fundamental equations of nature. This is an old and venerable area of study which goes 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 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 string theory, statistical mechanics, random processes, quantum informatics, and number theory. Many of the problems considered in the project have direct connections with these disciplines. Indeed, the concrete analytical questions addressed in the project are primarily motivated by the needs of the analytical theory of quantum entanglement - a phenomenon which is expected to play a key role in a practical realization of quantum computing, and by the challenges of analytical description of fundamental issue of phase transition in the quantum and statistical mechanics. The principal goal of this research project is to address various new analytical questions of the theory of integrable systems which have emerged from recent developments in random matrix theory and in the theory of exactly solvable quantum models. In this project, the PI proposes to focus on three directions of research: the asymptotic analysis of Toeplitz and Hankel determinants and the applications of these determinants in random matrices and statistical mechanics, the study of Fredholm determinants arising in random matrix theory, and the asymptotics of the correlation functions of non-free fermionic quantum spin models. 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.