Title: The Langlands Functoriality Conjecture: Problems in Number Theory and Representation Theory
Abstract: The study of L-functions forms an integral part of modern number theory. While this study dates back to Hecke in the 1920s, Robert Langlands, in the 1960s, began the study of L-functions from the point of view of automorphic representations. To an automorphic representation π of a reductive group G, Langlands associates an L-function, L(s, π ρ), where ρ is a representation of the so-called L-group associated to G. (In the simplest case, this is the group whose root datum is dual to that of G.) Roughly speaking, Langlands’ functoriality conjecture predicts that certain operations on the representations ρ of L-groups give rise to liftings of automorphic representations. In my talk, I will define these notions, give examples and motivation, and discuss recent work which addresses the functoriality conjecture and related questions in number theory.