Abstract

Symmetries play a fundamental role in mathematics.  I will begin by recalling two basic contexts in which symmetries occur:  symmetries of equations (Galois theory) and symmetries of harmonic analysis (discrete groups acting on symmetric spaces).  I will then outline how the Langlands program conjectures a framework in which these two contexts are fundamentally related.   While I will try to indicate some interesting contemporary results, my main focus will be on providing an introduction to the underlying ideas through simple but key examples.