Riemann-Hilbert problems, infinite matrices and their applications

Riemann-Hilbert问题(RHP)有着悠久而令人印象深刻的历史，可以追溯到Riemann的论文(1851年)和Hilbert在20世纪初的相关结果。Riemann-Hilbert问题与一维奇异积分算子、卷积算子、Toeplitz算子和Wiener-Hopf算子密切相关。Riemann-Hilbert问题可以描述为在给定的曲线上以给定的跳跃在复平面上寻找一个解析函数的问题。在这几种不同的形式中，这个问题吸引了许多著名的数学家。RHP的研究领域不断扩大，并在数学、数学物理甚至化学的许多应用领域找到了新的应用。目前RHP的重要意义在于它在随机矩阵理论、正交多项式和可积系统中的应用。可积偏微分方程组的Riemann-Hilbert方法起源于Manakov，Shabat和Zakharov于1975-1979年的工作，此后在孤子理论中得到了广泛的应用。关于量子精确可解模型的Riemann-Hilbert方法最近是在20世纪90年代由ITS、Izergin、柯尔平、斯拉夫诺夫、Deift和周的一系列工作中发展起来的。Riemann-Hilbert方法是由ITS、Fokas和Kitaev在1991年提出的关于正交多项式和矩阵模型的方法，它解决了与随机矩阵中的普适性有关的一些长期存在的问题。在形变理论中，当表征特征值的度量或局部化范围的参数变化时，随机矩阵模型与经典可积系统之间的关系就会显现出来。值得注意的是，由此产生的确定配分函数和关联函数的微分方程与可积系统理论中出现的某些方程属于同一类型。许多上述应用是通过某些类型的算子和无限矩阵，即Toeplitz矩阵和Hankel矩阵产生的，当矩阵的大小为无穷大时，需要研究它们的行列式的渐近性。Toeplitz矩阵和行列式的研究是由Otto Toeplitz在1907年开创的，目的是寻找希尔伯特泛函分析一般理论的具体例子。数学、物理和工程中的许多问题都可以用这些矩阵来表示；特别是在函数论、概率论、统计学和统计力学等领域，包括考夫曼和昂萨格在伊辛模型上的工作。

The Riemann-Hilbert problem (RHP) has a long and impressive history going back to Riemann's dissertation (1851) and Hilbert's related results at the beginning of the 20th century. The Riemann-Hilbert problem, which can be described as a problem of finding an analytic function in the complex plane with a prescribed jump across a given curve, is closely connected to one-dimensional singular integral operators, convolution operators, Toeplitz operators, and Wiener-Hopf operators. In these several different forms, the problem has attracted many famous mathematicians. The research area continues to expand rapidly and find new applications in many (applied) fields of mathematics, in (mathematical) physics and even in chemistry.A great deal of the current importance of the RHP is due to its use in random matrix theory, orthogonal polynomials and integrable systems. The Riemann-Hilbert method for integrable PDEs originated in the works of Manakov, Shabat, and Zakharov in 1975-1979, and since then it has been widely used in soliton theory. The Riemann-Hilbert approach to quantum exactly solvable models was most recently developed in the 1990s in the series of works by Its, Izergin, Korepin, Slavnov, Deift, and Zhou. The Riemann-Hilbert approach to orthogonal polynomials and matrix models was initiated in 1991 by Its, Fokas, and Kitaev, which has led to solving some of the long-standing problems in the asymptotics of orthogonal polynomials related to universalities in random matrices. The relations between random matrix models and classical integrable systems appear in deformation theory, when parameters characterizing the measures or the domain of localization of the eigenvalues are varied. The resulting differential equations determining the partition function and correlation functions are, remarkably, of the same type as certain equations appearing in the theory of integrable systems. They may be analyzed effectively through methods based on the RHP and by related approaches to the study of nonlinear asymptotics in the large N limit.Many of the aforementioned applications arise via certain classes of operators and infinite matrices, namely Toeplitz and Hankel matrices, and require the study of the asymptotics of their determinants when the size of the matrix goes to infinity. The study of Toeplitz matrices and determinants was initiated by Otto Toeplitz in 1907 to find concrete examples of Hilbert's general theory of functional analysis. A great variety of problems in mathematics, physics and engineering can be expressed in terms of these matrices; in particular in areas such as function theory, probability theory, statistics, and statistical mechanics, including Kaufman and Onsager's work on the Ising model.