Algebra/Topology seminar

Bryden Cais, Crystalline Galois representations

The main goal of algebraic number theory is to understand the automorphism group of the field of algebraic numbers, which can be thought of as the "fundamental group" of the integers.  This is an enormous and very complicated group, so to glimpse its inner mysteries, mathematicians study its continuous representations on finite dimensional vector spaces over topological fields.  As the fundamental group of the integers may be realized as an inverse limit of discrete groups, its natural topology is profinite; as a consequence, any continuous complex representation has finite image, and so is not very interesting!  In stark contrast, the study of such "Galois representations" over the p-adic numbers is both rich and rewarding.  In this talk, I will survey the exciting field of p-adic Galois representations, and discuss some new results regarding the behavior of "crystalline" p-adic representations under restriction to certain infinite-index subgroups of the (local at p) fundamental group. This is joint work with Tong Liu.