A quick introduction to the Langlands program

Abstract:  Questions about integer or rational solutions to polynomial equations (Diophantine problems) have occupied a central place in the study of number theory since its inception.  The Langlands program is a vast series of conjectures that can be applied to study Diophantine problems.  One tenet of the Langlands program is that certain arithmetic objects should be related to certain analytic objects. The modularity theorem of Andrew Wiles et al. is one such example of this correspondence. We will consider some of the early examples that influenced the formulation of the Langlands program and discuss the relationship with Diophantine equations.