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

黎曼-希尔伯特问题（Riemann-Hilbert problem，RHP）在许多应用中出现，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确定一个函数，该函数沿平面中的曲线沿着以规定的方式跳跃，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，后来由希尔伯特在世纪末提出。他们的研究一直处于纯数学和应用数学的前沿。直到最近，很少有人致力于实际计算这些问题的解决方案。该研究项目扩展了最近在进行数值研究方面的工作。预计RHP的解决方案将取得重大进展，从而增加对海啸、快速光通信和其他物理现象的了解，该项目的目标是开发新的计算工具，用于解决RHP及其扩展。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已连接到随机矩阵理论，非线性特殊功能，和非线性波动方程。RHP可以在黎曼曲面上提出，并且可以指定非线性跳跃条件。最近的发展涉及的研究人员和合作者已经导致了发展的准确和有效的数值算法的解决方案的RHP，黎曼曲面上提出的问题，并计算特殊的功能，如肖特基-克莱因素函数。然而，仍然存在许多悬而未决的问题，特别是在新的应用方面。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有效算法，并将其部署在应用程序中。研究人员，博士后学者和合作者汇集了成功开展合作项目所需的不同领域的独特专业知识组合。

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.