Some Analytical Aspects of the Theory of Integrable Systems

可积系统理论的一些分析方面

• 批准号: 1361856
• 负责人: Alexander Its
• 金额: $ 34.2万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361856&HistoricalAwards=false
• 关键词: Some; Analytical; Aspects; Theory; Integrable

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.

所研究的方程类别称为集成系统，包括多种自然的基本方程。这是一个古老而古老的学习领域，可以追溯到Liouville，Gauss和Poincare的古典作品。当前，综合系统的理论已成为一个不断扩展的领域，它是现代数学和理论物理学许多分支的新思想的主要来源之一，起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产生的一些基本数学模型提供了有效的分析工具。除了传统的微分方程领域外，可集成的技术在诸如字符串理论，统计力学，随机过程，量子信息学和数字理论等各个领域中变得普遍。项目中考虑的许多问题与这些学科有直接的联系。实际上，项目中解决的具体分析问题主要是由量子纠缠分析理论的需求引起的 - 这种现象有望在实现量子计算的实际实现中起关键作用，以及量子过渡和统计机制中相位过渡问题的分析描述的挑战。该研究项目的主要目的是解决对整合系统理论的各种新分析问题，这些问题来自随机矩阵理论的最新发展以及确切可解决的量子模型的理论。在该项目中，PI提议专注于三个研究方向：对Toeplitz和Hankel决定因素的渐近分析，以及这些决定因素在随机矩阵和统计力学中的应用，在随机基质理论中引起的Fredholm决定因素的研究，以及相关的非元素质量模型的相关功能。 这些方向中的每一个都由一系列具体问题表示，并且将在同一分析框架中进行研究，即Riemann-Hilbert方法。