Collaborative Research: Riemann-Hilbert Problems and Riemann Surfaces: Computations and Applications

黎曼-希尔伯特问题（RHPs）出现在大量的应用中，从描述海啸的方程到对核能的理解。在其最基本的形式中，RHP确定了一个函数，该函数沿着平面上的曲线以规定的方式跳跃，并且在远离跳跃发生的地方具有指定的行为。这些问题首先由黎曼提出，后来由希尔伯特在19世纪末提出。他们的研究一直走在纯数学和应用数学的前沿。直到最近，对这类问题的解的实际计算投入的努力还很少。这个研究项目扩展了最近在进行数值研究方面的工作。预计在解决RHPs方面将取得重大进展，从而增加对海啸、快速光通信和其他物理现象的了解。该项目的目标是为rhp及其扩展的解决方案开发新的计算工具。传统上，rhp出现在奇异积分方程和Wiener-Hopf技术的背景下。最近，rhp已与随机矩阵理论、非线性特殊函数和非线性波动方程联系起来。rhp可以在Riemann曲面上提出，并且可以指定非线性跳跃条件。研究人员和合作者最近的发展导致了rhp解的精确和有效的数值算法的发展，用于黎曼曲面上的问题，以及特殊函数（如肖特基-克莱因素函数）的计算。然而，许多尚未解决的问题仍然存在，特别是涉及到新的应用程序。该项目旨在开发新的计算方法来解决这些问题，重点是开发能够处理复杂几何形状的快速高效算法，并将其部署到应用中。研究人员、博士后学者和合作者将成功开展合作项目所需的不同领域的专业知识结合在一起。

Riemann-Hilbert problems (RHPs) arise in a plethora of applications, varying from equations describing tsunamis to the understanding of nuclear energy. In its most basic form, a RHP determines a function that jumps in a prescribed way along a curve in the plane and has specified behavior far away from where the jump occurs. Such problems were first posed by Riemann and later by Hilbert at the end of the 19th century. Their study has been at the forefront of pure and applied mathematics. Until recently, little effort had been devoted to the actual computation of solutions of such problems. This research project extends recent work in carrying out numerical investigations. It is anticipated that major advances will be made in the solution of RHPs, allowing for the increased understanding of tsunamis, fast optical communication, and other physical phenomena.The goal of the project is to develop new computational tools for the solution of RHPs and their extensions. Traditionally, RHPs arise in the context of singular integral equations and the Wiener-Hopf technique. More recently, RHPs have been connected to random matrix theory, nonlinear special functions, and nonlinear wave equations. RHPs may be posed on Riemann surfaces, and nonlinear jump conditions may be specified. Recent developments involving the investigators and collaborators have led to the development of accurate and efficient numerical algorithms for the solution of RHPs, for problems posed on Riemann surfaces, and for the computation of special functions such as the Schottky-Klein prime function. However, many open problems remain, particularly concerning new applications. This project aims to develop new computational methods to solve these problems, with an emphasis on the development of fast and efficient algorithms that can deal with complicated geometries, and to deploy them in applications. The investigators, postdoctoral scholar, and collaborators bring together a unique combination of expertise in the different areas needed to successfully carry out the collaborative project.