Riemann-Hilbert problem and integrable systems

摘要本课题是研究Riemann-Hilbert问题方法的各种应用。这些应用包括非线性薛定谔方程的摄动理论和与随机矩阵理论有关的几个新发展的领域。一本关于黎曼-希尔伯特问题的书也包括在内。近年来，黎曼-希尔伯特问题在数学的几个重要领域被发现。前面提到的一些应用也具有跨学科的影响。例如，随机矩阵理论的主要应用是在物理学中，对某些微扰微分方程的研究增强了对光纤中信号传输的理解。

PI: Xin Zhou, Duke UniversityDMS-0300844AbstractThis project is to study various applications of the Riemann-Hilbert problem method. These applications include the perturbation theory of nonlinear Schroedinger equations and severalnewly developed areas in connection with the theory of random matrices. A book project on Riemann-Hilbert problems is also included.Riemann-Hilbert problems have been found in several important areas in mathematics in recent years. Some afore mentioned applications also have interdisciplinary impact. For example, themain application of the theory of random matrices is in physics and the study of certain perturbed differential equations enhances the understanding of signal transmissions in fibre optics.