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: 1001777
• 负责人: Alexander Its
• 金额: $ 17.8万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2014-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001777&HistoricalAwards=false
• 关键词: Some; Analytical; Aspects; Theory; Integrable

The principal goal of the research project is to address some of the new analytical questions of the theory of integrable systems which emerged from the recent developments in random matrix theory and in the related areas of exactly solvable quantum models. The problems under consideration include the analytical description of the Painlev\'e - type universalities in random matrix theory and the related study of special families of Painlev\'e transcendents and generalized Painlev\'e transcendents arising in the critical asymptotics in random matrices, statistical mechanics and in the enumerative topology. The third direction of the project is concerned with the analytical investigation of the crossover phenomena in the classical theory of Toeplitz and Hankel determinants, i.e. with the study of the Painlev\'e-type transition behavior between different non-critical asymptotic regimes exhibited by large size Toeplitz and Hankel determinants. Each of the above mentioned directions is represented by a collection of concrete problems, and they are proposed to be investigated within the same analytical framework, which is the Riemann-Hilbert method.The theory of integrable systems is an expanding area which plays anincreasingly 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 study of some of the fundamental mathematical models arising in modern nonlinear science and technology. The new areas where the analytic techniques of integrable systems become more and more common include random matrices, quantum field and string theories, enumerative topology, stochastic processes, number theory, and, most recently, entanglement in quantum chain systems which is expected to play prominent role in future quantum computing technology. The problems considered in the proposal have direct connections with the mentioned disciplines. In particular, part of the proposal related to the Toeplitz and Hankel determinants is primary motivated by the needs of the analytical theory of quantum entanglement and by the challenges of analytical description of critical behavior in important stochastic and quantum field models. This part of the proposal is alsoessential for testing the random matrix predictions in number theory. Proposed study of the Painlev\'e - type universalities and the families of special Painlev\'e functions has direct relation to enumerative topology and string theory. Success in achievement of the proposal's goals will have a notable impact on research in all these areas.

研究项目的主要目的是解决综合系统理论的一些新分析问题，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相关领域。所考虑的问题包括对随机矩阵理论中Painlev \'e-类型普遍性的分析描述，以及对Painlev \ e超验和广义的特殊家庭的相关研究，以及在随机矩阵，统计机制和涉嫌拓扑中引起的关键渐近物质中引起的临界差异。该项目的第三个方向与Toeplitz和Hankel决定因素的经典理论进行了对跨界现象的分析研究，即，通过研究大型Toeplitz和Hankel确定的不同非临界渐近性方案之间的Painlev \'e-e-type过渡行为。 上述每个方向都由一系列具体问题表示，并建议在相同的分析框架中进行调查，这是Riemann-Hilbert方法。整合系统的理论是一个不断扩展的领域，它扮演着一个非常重要的角色，它是新分析和对现代数学的新分析和代价构想的主要来源之一，是现代数学的主要构想的主要来源之一。同时，它提供了一种有效的分析工具，用于研究现代非线性科学和技术中产生的一些基本数学模型。可集成系统的分析技术变得越来越普遍的新领域包括随机矩阵，量子场和弦理论，枚举拓扑，随机过程，数量理论以及最近在量子链系统中的纠缠，这些量子系统有望在未来的量子计算技术中发挥着重要作用。提案中考虑的问题与上述学科有直接联系。特别是，与Toeplitz和Hankel决定因素相关的一部分提案是由量子纠缠分析理论的需求以及重要随机和量子场模型中关键行为的分析描述的挑战的主要动机。该提案的这一部分也是针对数字理论中随机矩阵预测的必不可少的。 建议对Painlev \'e-普遍性的研究和特殊的Painlev'e功能的家族与枚举拓扑和弦理论直接关系。实现提案目标的成功将对所有这些领域的研究产生显着影响。