Galois Representations, Monodromy Groups, and Motives

在整个数学科学中，对称是一个强大的组织原则。在研究整数，特别是质数的数论中，没有更直观的几何设置（90度旋转下正方形的对称等）所遇到的那种明显的对称性。尽管如此，我们对素数最深刻的认识大多来自于它们与多项式方程的“对称性”之间的联系。这个项目专注于伽罗瓦表示的研究，伽罗瓦表示是关于素数信息的自然包。由罗伯特·朗兰兹（Robert Langlands）发起的现代数论的核心项目之一，提出了通过将伽罗瓦表示与几何中出现的截然不同的数学对象联系起来，来“解开”伽罗瓦表示的新方法。虽然它们本质上是关于质数的，但这些关系甚至在基础物理学中产生了反响。这个项目既包括构建新的伽罗瓦表示，也包括在朗兰兹的猜想中解开它们所包含的信息。这个项目包括两个研究项目。第一部分将研究任意约化群中的伽罗瓦表示的变形，以尽可能广泛地建立给定模p表示的几何p进变形的存在性。这项工作很可能为将Serre的模块化猜想推广到一般约化群提供步骤。这种几何p进表示被期望是（半简单）伽罗瓦表示，出现在数域上代数变异的上同调中。第二个项目旨在研究Fontaine-Mazur类型的一些问题，寻求建立某些伽罗瓦表示的动机起源，或者在更几何的环境中，曲线上的局部系统。这项工作将需要详细阐述几何朗兰兹纲领的基本对象的动机结构，即使对于算术朗兰兹纲领来说，这个问题也应该是一个长期独立的问题。

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.