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，尤其是素数的数论中，没有在更直观的几何环境中遇到的那种明显的对称性(正方形在90度旋转下的对称性，等等)。尽管如此，我们对素数最深入的了解大多来自于它们与多项式方程的“对称性”的联系。这个项目专注于伽罗瓦表示的研究，这是关于素数的信息的自然包。现代数论的核心程序之一，由罗伯特·朗兰兹发起，提出了通过将伽罗瓦表示与几何学中出现的显著不同的数学对象相关联来“拆解”伽罗瓦表示的新方法。尽管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。这个项目包括构造新的伽罗瓦表示和根据朗兰兹猜想解开其中包含的信息。这个项目包括两个研究项目。第一部分将研究赋值于任意约化群中的Galois表示的变形，以期在尽可能一般的情况下建立给定mod p表示的几何p进变形的存在性。这项工作可能会把Serre的模性猜想推广到一般的约化群。这样的几何p-进表示被认为是数域上代数簇上同调中的(半单)Galois表示。第二个程序旨在研究Fontaine-Mazur类型的一些问题，试图建立某些伽罗瓦表示的动机起源，或者在更几何的背景下，建立曲线上的局部系统。这项工作将需要详细阐述几何朗兰兹计划基本目标背后的动机结构，即使对算术朗兰兹计划来说，这也是一个长期独立感兴趣的问题。

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.