Mathematical Sciences: Topics in Analysis

萨纳克教授的研究方向将是自同构形式的谱理论，以及在数论、组合学和数学物理中的应用。他计划在这一领域研究的两个悬而未决的问题是(1)非均匀格子的Maass尖点形式的存在性和(2)“听鼓的形状”问题，即关于等谱流形集可以说什么。奥斯古德教授将研究几何函数论中的几个问题，其中一些是与萨纳克共同研究的。本文将研究经典Schwarzian导数推广到高维曲面的等谱问题。这项研究将涉及几何学和分析的结合，以及在其他数学领域的应用。事实证明，自同构形理论是一门起源于数论的学科，通常是作为算术函数的母函数而产生的，在其他领域也有很多应用。这一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.