Some Analytical Aspects of the Theory of Integrable Systems

该研究项目的主要目标是解决可积系统理论的一些新的分析问题，这些问题出现在随机矩阵理论和相关领域的精确可解量子模型的最新发展中。所考虑的问题包括随机矩阵理论中Painlev‘e型普适性的解析描述，以及随机矩阵、统计力学和计数拓扑中临界渐近中出现的特殊Painlev’e超越族和广义Painlev‘e超越族的相关研究。项目的第三个方向是关于经典Toeplitz和Hankel行列式理论中交叉现象的分析研究，即研究大尺寸Toeplitz和Hankel行列式所表现出的不同非临界渐近区域之间的Painlev‘e型转变行为。上述每个方向都由一组具体的问题表示，并建议在同一分析框架内进行研究，这就是Riemann-Hilbert方法。可积系统理论是一个不断扩展的领域，它作为现代数学和理论物理的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个有效的分析工具。可积系统的分析技术越来越普遍的新领域包括随机矩阵、量子场和弦理论、计数拓扑、随机过程、数论，以及最近有望在未来的量子计算技术中发挥重要作用的量子链系统中的纠缠。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行列式有关的部分建议主要是出于量子纠缠分析理论的需要，以及在重要的随机场和量子场模型中对临界行为的解析描述的挑战。该建议的这一部分对于检验数论中的随机矩阵预测也是必不可少的。Painlev-e型普适性和特殊Painlev函数族的研究与列举拓扑和弦理论有直接关系。成功实现该提案的目标将对所有这些领域的研究产生显著影响。

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.