Explicit Methods for the Local Langlands Correspondence

这个项目植根于数学的两个古老部分：表示论（对称性的研究）和数论（方程的数值解）。虽然这似乎是两个完全不同的数学领域，但局部朗兰兹通信（LLC）预测了它们之间令人惊讶的关系。粗略地说，它说无限维对称性应该对应于方程，方程的解具有有限维对称性的相关集合。这个项目的目标是首先，发现和明确验证新的和有趣的例子的LLC，第二，使用的预测LLC作出新的发现数论和表示理论.这个项目是关于表示理论和数论之间的相互作用，一般约化群的局部朗兰兹对应预测. LLC是局部伽罗瓦群的表示理论与p-adic域上的约化群的表示理论之间的预测关系。这个项目的目标是通过明确的方法发现和理解LLC的新的和有趣的方面。有五个研究方向：伽罗瓦群到复李群的李本原表示;参数为李本原的p-adic群的表示;将已知的上层表示结构扩展到更高的深度;天鹅导体的不等式;深度零表示和LLC的约旦分解。

This project is rooted in two ancient parts of mathematics: representation theory (the study of symmetry) and number theory (numerical solutions of equations). Though these appear to be two quite different areas of mathematics, the Local Langlands Correspondence (LLC) predicts surprising relations between them. Roughly speaking, it says that infinite-dimensional symmetries should correspond to equations whose solutions have a related collection of finite-dimensional symmetries. The goals of this project are first, to discover and explicitly verify new and interesting examples of the LLC, and second, to use the predictions of the LLC to make new discoveries in number theory and representation theory.This project is about interactions between representation theory and number theory, as predicted by the Local Langlands Correspondence for general reductive groups. The LLC is a body of predicted relations between the representation theory of local Galois groups to representation theory of reductive groups over a p-adic field. The goals in this project are to discover and understand new and interesting aspects of the LLC, via explicit methods. There are five proposed directions of inquiry: Lie-primitive representations of Galois groups into complex Lie groups; representations of p-adic groups whose parameters are Lie-primitive; extending known constructions of epipelagic representations to higher depth; inequalities for Swan conductors; and Jordan decomposition of depth zero representations and LLC.