Groups, Arithmetic and Monodromy

该提议的主要目标是更好地理解伽罗瓦表示的图像。通常，这样的像看起来像约化群方案的l进点群，并且人们希望在有利的情况下证明这一点（例如，对于函数域上的光滑射影变化）。在一个不同的方向上，提议者打算研究伽罗瓦表示在多大程度上可以用于解决有理数域上Lie型群的反伽罗瓦问题。其他目标包括研究有限群作用的商的几何，以研究阿贝尔变上的伽罗瓦模结构；分析不需要约化的群的渐近不变理论，着眼于计算单群。总的来说，申请者打算研究群论（对称的抽象研究）、代数数论（满足有理系数多项式方程的数系统的研究）和代数几何（若干变量多项式方程系统的几何研究）中思想的相互作用。所提出的工作的中心主题是单一性，从概念上讲，它是对与一个闭合回路的点相关的几何物体所揭示的对称性的研究。这个项目包括提高我们对群论的理解，群论既是目的本身，也是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.