Groups, Arithmetic and Monodromy

该提案的主要目标是更好地理解伽罗瓦表示的图像。 通常这样的图像看起来像一个约化群方案的l-adic点群，人们希望在有利的情况下证明这一点（例如，函数域上的光滑射影簇）。 在另一个方向上，提议者打算研究伽罗瓦表示在多大程度上可以用来解决有理数域上李群的逆伽罗瓦问题。 该计划的其他目标包括研究有限群作用的等价物的几何，以研究交换簇上的伽罗瓦模结构，以及分析不需要是约化的群的渐近不变量理论，着眼于计算单值群。一般而言，建议者打算研究群论中思想的相互作用（对称的抽象研究），代数数论（研究满足有理系数多项式方程的数字系统）和代数几何（研究多变量多项式方程系统的几何）。 一个中心主题的拟议工作是monodromy，这是，从概念上讲，研究的对称性所揭示的一个几何对象相关联的一个点，其中遵循一个封闭的循环。 该项目涉及提高我们对群论的理解，既作为目的本身，也作为研究数域对称性的工具。

The primary goal of the proposal is to better understand images of Galois representations. Typically such an image looks like the group of l-adic points of a reductive group scheme, and one would like to prove this in favorable situations (for example, for smooth projective varieties over function fields). In a different direction the proposer intends to investigate how far Galois representations can be used to solve the inverse Galois problem for groups of Lie type over the field of rational numbers. Other goals of the proposal include studying the geometry of quotients of finite group actions in order to investigate Galois module structure on abelian varieties and analyzing the asymptotic invariant-theory of groups which need not be reductive, with an eye to computing monodromy groups.In general terms, the proposer intends to investigate the interplay of ideas in group theory (the abstract study of symmetry), algebraic number theory (the study of systems of numbers satisfying polynomial equations with rational coefficients), and algebraic geometry (the geometric study of systems of polynomial equations in several variables). A central theme of the proposed work is monodromy, which is, conceptually, the study of the symmetries revealed by a geometric object associated to a point which follows a closed loop. The project involves improving our understanding of group theory both as an end in itself and as a tool to study symmetries of number fields.