Shimura varieties and the search for a Langlands transform

The Langlands reciprocity conjectures predict the existence of a correspondence between certain classes of representations of Galois groups of number fields and automorphic representations. The study of the geometry of Shimura varieties has been central in establishing these conjectures in the cases where they are known. The talk will explain how Shimura varieties provide a link between automorphic forms and Galois theory, and will review some of the most recent results.