On Basic Structures of Automorphic Representations

(Abstract)

We start with the multiplicative structure of numbers and the elementary local-global structure of numbers, and then discuss the local-global structure in elementary harmonic analysis and how to use them to study objects in number theory, like the Riemann zeta functions. The modern theory of automorphic forms is a natural extension of these classical theory, which encodes the deep local-global relations for harmonic analysis or representations of reductive algebraic groups over number fields. This becomes the main part of the Langlands program. I will discuss my recent work along these lines of ideas