Abstract: The natural analog for a groupoid of the representation of a group on a vector space is a representation on a vector bundles or, with a little more structure, sheaves. This talk will introduce the representation theory of groupoids, and its relation to Morita equivalence.

Abstract: This talk will discuss the local splitting behaviour of ordinary modular Galois representations and relate them to companion forms and complex multiplication. Two modular forms (specifically $p$-ordinary, normalized eigenforms) are said to be "companions" if the Galois representations attached to them satisfy a certain congruence property. Companion forms modulo $p$ play a role in the weight optimization part of (the recently established) Serre's Modularity Conjecture. Companion forms modulo $p^n$ can be used to reformulate a question of Greenberg about when a normalized eigenform has CM (complex multiplication).