Mathematical Sciences: Studies of Arithmetic Automorphic Forms

自守表示的领域是一个交汇点 数学的各个分支--例如，数论， 李理论，代数几何--以及数学物理。 它的起源是在尝试应用的技术， 群的无限维表示理论，以获得 更深入地理解L-级数，一个数的主导概念 理论 作为表示论的一个子集， 在过去的几十年里。 因此，现在人们发现， 技术和应用的显着扩展， 模形式的分析理论，迹公式， 代数几何、组合数学和泛函分析。 弗里克教授的研究涉及 两个群之间的自守表示。 最近 协同工作建立了元对应关系， 关于一般线性群上的自守形式， 任意拓扑覆盖 这项工作开发了新的 技术，同时也简化了计算。 各种 这些结果的应用包括弗里克教授的 后续工作。 目前的计划要求在几个 方向：提升定理自守形式。 转让 稳定轨道积分 轨道积分的匹配 平滑函数 字符和扭曲的计算 字符. 追踪公式。 GL（n）上的Ramanujan猜想 函数字段。 同余关系。 的应用 关于Iwahori子群的Hecke代数

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.