Mathematical Sciences: Spectra and Geometry: Random Media and Hyperbolic Manifolds

Hislop将继续他的研究的相互作用， 几何形状和光谱性质。 项目的第一部分 涉及随机介质的光谱和传输特性。 他 我将研究安德森本地化的一个家庭的连续 随机哈密顿量和无序状态下的麦克斯韦方程 媒体 该项目的第二部分将集中于 无限体积双曲流形 他将完成 Eisenstein级数和S-矩阵的特征 几何有限双曲流形 现代物理学，量子力学和相对论，是一个 二十世纪的产物。 它牢固地建立在 上世纪人们试图研究物质的微观结构 并掌握远距离行动的概念， 电磁和热辐射。 数学 这些发展的基础，统称为 数学物理，范围从详细分析 薛定谔算子，控制着粒子的动力学， 统一场论，试图统一四个已知的 形成一个单一的理论。

Hislop will continue his study of the interplay of effective geometry and spectral properties. The first part of the project concerns spectral and transport properties of random media. He will investigate Anderson localization for a family of continuous random Hamiltonians and for Maxwell's equations in disordered media. The second part of the project will concentrate on infinite volume hyperbolic manifolds. He will complete the characterization of the Eisenstein series and S-matrix for geometrically finite hyperbolic manifolds. Modern physics, quantum mechanics and relativity, is a product of the twentieth century. It is founded firmly in the last century's attempt to address the microstructure of matter and to come to grips with the concepts of action-at-a distance, electro-magnetism, and heat radiation. The mathematical foundations for these developments, collectively called mathematical physics, ranges from detailed analysis of Schroedinger operators, which governs the dynamics of particles, to unified field theory, which attempts to unite the four known forces into a single theory.