Locally Symmetric Spaces Seminar

It is well known that an irreducible representation of a group $G$ over a field $k$ admits a universal deformation to a representation over a complete Noetherian local ring, provided that it is absolutely irreducible, i.e. remains irreducible after extending scalars to an algebraic closure of $k$, and satisfies a certain cohomological finiteness condition. I will discuss an enhancement of this construction: a universal deformation over a simplicial commutative ring $R$. The ring $\pi_0(R)$ is canonically isomorphic to the classical universal deformation ring, but $R$ may have interesting non-trivial higher homotopy. My talk will introduce this ring and the deformation functor it represents. I will also explain how to study its structure using cohomology, and its behavior under changing $G$ (e.g. changing ramification data for Galois groups). This is joint work with Akshay Venkatesh.