Mathematical Sciences: Topics in Analysis

萨纳克教授的研究将在光谱理论 自守形式与数论的应用 组合学和数学物理学。 其中两个悬而未决的问题 他计划在这一领域开展工作的问题是：（1） 非均匀格的Maass尖点型的存在性;（2）非均匀格的Maass尖点型 “听鼓形”问题，即可以说什么 关于流形的等谱集 奥斯古德教授会 工作的几个问题，在几何函数理论，一些在 与Sarnak合作。 这项工作将涉及等光谱 问题的曲面与推广的经典 Schwarzian导数到高维。 这项研究将涉及几何和 分析与应用到其他领域的数学。 它 自守形式理论，一个由 最初来自数论，通常作为生成函数， 算术函数，在其他领域有许多应用。 这 理论是本研究的重点之一。

Professor Sarnak's research will be in the spectral theory of automorphic forms with applications to number theory, combinatorics and mathematical physics. Two of the unsolved problems in this area which he plans to work on are (1) The existence of Maass cusp forms for nonuniform lattices and (2) The "hearing the shape of a drum" problem, i.e. what can be said about an isospectral set of manifolds. Professor Osgood will work on several problems in geometric function theory, some in joint work with Sarnak. This work will concern the isospectral problem for surfaces with a generalization of the classical Schwarzian derivative to higher dimensions. This research will involve a combination of geometry and analysis with applications to other areas of mathematics. It turns out that the theory of automorphic forms, a subject arising originally from number theory often as generating functions for arithmetic functions, has many applications to other areas. This theory is one of the main focuses of this research.