Galois Representations, Monodromy Groups, and Motives

在整个数学科学中，对称性是一个强有力的组织原则。在研究整数，特别是素数的数论中，没有在更直观的几何设置中遇到的那种明显的对称性（90度旋转下的正方形的对称性等）。尽管如此，我们对素数的最深层次的了解大多来自于它们与多项式方程的“对称性”的联系。这个项目的重点是伽罗瓦表示的研究，这是自然的包，这些信息的素数。由罗伯特·朗兰兹（Robert Langlands）发起的现代数论的核心程序之一，提出了将伽罗瓦表示与几何中出现的显著不同的数学对象联系起来的新方法来“解包”伽罗瓦表示。虽然它们本质上是关于素数的，但这些关系甚至在基础物理学中也有反响。这个计画包括建构新的伽罗瓦表示，以及根据朗兰兹的理论解压缩它们所包含的资讯。第一个将研究变形的伽罗瓦表示值在任意约化群，以期建立尽可能大的一般性存在的几何p-adic变形的一个给定的mod p表示。 这项工作可能会为将塞尔的模块性猜想推广到一般约化群提供步骤。这样的几何p-adic表示被认为是数域上代数簇的上同调中出现的（半简单）伽罗瓦表示。第二个项目旨在研究Fontaine-Mazur类型的一些问题，寻求建立某些Galois表示的motivic起源，或者在更几何的设置中，局部系统在曲线上。这项工作将需要详细说明几何朗兰兹程序的基本对象的motivic结构，即使对于算术朗兰兹程序来说，这个问题也应该是长期独立感兴趣的。

Throughout the mathematical sciences, symmetry is a powerful organizing principle. In number theory, which studies the integers, especially the prime numbers, there are no manifest symmetries of the kind encountered in more intuitive, geometric settings (symmetry of a square under 90-degree rotations, etc.). Nevertheless, much of our deepest knowledge of the primes comes from their connections with the "symmetries" of polynomial equations. This project focuses on the study of Galois representations, which are natural packages for this information about the prime numbers. One of the central programs of modern number theory, initiated by Robert Langlands, proposes new ways to "unpack" Galois representations by relating them to remarkably different mathematical objects arising in geometry. Although they are essentially about the prime numbers, these relations have even had reverberations in fundamental physics. This project involves both constructing new Galois representations and unpacking the information they contain in the light of Langlands' conjectures.This project comprises two research programs. The first will study deformations of Galois representations valued in arbitrary reductive groups, with a view toward establishing in as great generality as possible the existence of geometric p-adic deformations of a given mod p representation. The work is likely to furnish steps toward generalizations of Serre's modularity conjecture to general reductive groups. Such geometric p-adic representations are expected to be the (semi-simple) Galois representations occurring in the cohomology of algebraic varieties over number fields. The second program aims to study a number of questions of Fontaine-Mazur type, seeking to establish the motivic origin of certain Galois representations or, in a more geometric setting, local systems over a curve. The work will require elaborating the motivic structures underlying basic objects of the geometric Langlands program, a problem that should be of long-term independent interest even for the arithmetic Langlands program.