Mathematical Sciences: Studies in the Analytic Theory of Automorphic Forms

该奖项支持马里兰大学帕克分校的Isaac Efrat教授在自同构形式分析理论方面的研究。Efrat博士提议研究作为秩一模离散群空间商的图和面的调和分析中的一系列问题。这些问题源于这些群的离散谱的存在性问题，以及它们的离散特征函数的构造问题。Efrat教授项目的其他方面包括对非同余子群上某些算术定义的爱森斯坦级数的研究。非欧几里得平面几何开始于19世纪初，当时只是数学上的一种好奇，但到19世纪末，数学家们已经意识到，许多具有根本重要性的物体在其基本性质上都是非欧几里得的。对非欧几里得平面几何的详细研究产生了现代数学的几个分支，其中模和自同构形式的研究是最活跃的分支之一。这个领域主要关注关于整数的问题，但在使用几何和分析时，它保留了与历史根源的联系。

This award supports the research in the analytic theory of automorphic forms of Professor Isaac Efrat of the University of Maryland at College Park. Dr. Efrat has proposed to investigate a collection of problems in the harmonic analysis of graphs and surfaces that arise as quotients of spaces of rank one, modulo discrete groups. These problems stem from questions about the existence of discrete spectra for such groups, and the construc- tion of discrete eigenfunctions for them. Other aspects of Professor Efrat's project involve the study of certain arith- metically defined Eisenstein series attached to non-congruence subgroups. Non-Euclidean plane geometry began in the early nineteenth century as a mathematical curiosity, but by the end of that century, mathematicians had realized that many objects of fundamental importance are non-Euclidean in their basic nature. The detailed study of non-Euclidean plane geometries has given rise to several branches of modern mathematics, of which the study of modular and automorphic forms is one of the most active. This field is principally concerned with questions about the whole numbers, but in its use of geometry and analysis, it retains connection to its historical roots.