This article is firstly a historic review of the theory of Riemann–Hilbert problems with particular emphasis placed on their original appearance in the context of Hilbert’s 21st problem and Plemelj’s work associated with it. The secondary purpose of this note is to invite a new generation of mathematicians to the fascinating world of Riemann–Hilbert techniques and their modern appearances in nonlinear mathematical physics. We set out to achieve this goal with six examples, including a new proof of the integro-differential Painlev-II formula of Amir etal (2011 Commun. Pure Appl. Math. 64 466–537) that enters in the description of the Kardar–Parisi–Zhang crossover distribution. Parts of this text are based on the author’s Szegő prize lecture at the 15th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA) in Hagenberg, Austria.

这篇文章首先是对黎曼-希尔伯特问题理论的历史回顾，特别强调了它们在希尔伯特第 21 个问题和普莱梅利相关工作的背景下的最初出现。本笔记的第二个目的是邀请新一代数学家进入 Riemann-Hilbert 技术的迷人世界及其在非线性数学物理中的现代表现。我们着手通过六个例子来实现这一目标，包括 Amir等人(2011 Commun. Pure Appl. Math. 64)的积分微分 Painlev-II 公式的新证明466–537) 进入 Kardar-Parisi-Zhang 交叉分布的描述。本文的部分内容基于作者在奥地利哈根贝格举行的第 15 届正交多项式、特殊函数和应用国际研讨会 (OPSFA) 上的 Szegő 奖演讲。