Mathematical Sciences: Scattering Theory

这个项目的目标是分析数论、函数论和散射论中的某些问题。所使用的技术涉及拉普拉斯-贝尔特拉米算子作用于有限边数和有限或无限体积的基本区域上的自同构函数的谱理论。拉普拉斯-贝尔特拉米算符是这三个领域的核心，代表了一个值得集中研究的重要概念。这项工作以及最近完成的关于显式扩展器和Ramanujan图的工作给出了可能的最佳扩展器；这些是交换网络以及排序方案的构建块。给出了射影平面上的Moser不等式的一个锐化形式，并给出了有限体积曲面上在拉普拉斯(以及散射矩阵行列式的缠绕数)扰动下尖点形式破坏的详细估计。

The goals of this project are the analysis of certain problems in number theory, function theory and scattering theory. Techniques to be used involve spectral theory of the Laplace-Beltrami operator acting on automorphic functions over a fundamental domain with a finite number of sides and of either finite or infinite volume. The Laplace-Beltrami operator is central to all three fields representing an important concept worthy of concentrated study. This work as well as that recently completed on explicit expanders and Ramanujan graphs gives the best possible expanders; these are the building blocks for switching networks as well as sorting schemes. A sharpened form of the Moser inequality for the projective plane and detailed estimates on the destruction of cusp forms under perturbation of the Laplacian (and thus the winding number of the determinant of the scattering matrix) on surfaces of finite volume are expected.