Mathematical Sciences: Scattering Theory

该项目的目标是分析某些 数论、函数论和散射问题 理论 要使用的技术涉及光谱理论的 Laplace-Beltrami算子作用于环上的自守函数 一个基本域，有有限条边， 体积有限或无限。 拉普拉斯-贝尔特拉米 运算符是所有三个字段的中心， 值得重点研究的重要概念。 这项工作以及最近完成的显式 扩展器和Ramanujan图给出了最好的可能 扩展器;这些是交换网络的构建块 以及分类方案。 一种磨尖了的莫泽刀 射影平面的不等式和关于 尖点形式在扰动下的破坏 Laplacian（因此， 散射矩阵）是 预期

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.