Mathematical Sciences: Studies of Arithmetic Automorphic Forms

自同构表示的领域是数学的各个分支--例如，数论、李理论、代数几何--和数学物理的交汇点。它的起源是试图应用群的无穷维表示论的技巧来加深对数论中占主导地位的概念L级数的理解。作为表征理论的一个子集，在过去的几十年里取得了很大的进展。因此，现在人们发现了涉及模形式分析理论、迹公式、代数几何技术、组合学和泛函分析的大量技术和应用。Flicker教授的研究涉及两个群之间的自同构表征的迁移。最近的合作工作建立了一般线性群上的自同构形式与任意拓扑覆盖之间的亚可解对应。这项工作开发了新的技术，同时也简化了计算。这些结果的各种应用构成了Flicker教授随后的工作。目前的计划需要在几个方向上进行工作：自同构形的提升定理。稳定轨道积分的转移。光滑函数轨道积分的匹配。字符和扭曲字符的计算。跟踪公式。函数域上GL(N)的Ramanujan猜想同余关系。Hecke代数关于Iwahori子群的应用。

The field of automorphic representations is a meeting ground of various branches of mathematics -- for example, number theory, Lie theory, algebraic geometry -- and of mathematical physics. Its origins are found in the attempt to apply the techniques of infinite-dimensional representation theory of groups to obtain a deeper understanding of L-series, a dominating concept in number theory. As a subset of representation theory, great progress has been made in the past few decades. Thus, one now finds a remarkable expanse of technique and applications involving the analytic theory of modular forms, trace formulas, techniques of algebraic geometry, combinatorics, and functional analysis. The research of Professor Flicker deals with the transfer of automorphic representations between two groups. Recent collaborative work establishes the metaplectic correspondence relating automorphic forms on the general linear group and an arbitrary topological covering. This work develops new techniques while also simplifying computations. Various applications of these results comprise Professor Flicker's subsequent work. Current plans call for work in several directions: Lifting theorems for automorphic forms. Transfer of stable orbital integrals. Matching of orbital integrals of smooth functions. Computations of characters and twisted characters. Trace formulas. Ramanujan conjecture for GL(n) over a function field. Congruence relations. Applications of the Hecke algebra with respect to an Iwahori subgroup.