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.
