Langlands proposed an extraordinary correspondence between representations of Galois groups and automorphic forms, which has deep, and completely unexpected, implications for the study of both objects. The simplest special case is Gauss' law of quadratic reciprocity. In the so called `regular, self-dual' case much progress has been made in the roughly 40 years since Langlands made these conjectures. In these talks I will discuss recent progress in regular, but non-self-dual case. In this case the automorphic forms in question can be realized as cohomology classes for arithmetic locally symmetric spaces, i.e., quotients of symmetric spaces by discrete groups. Thus instead of the Langlands correspondence being a relationship between algebra and analysis, it can be thought of as a relationship between algebra and topology. This realization of the Langlands correspondence is in many ways more concrete. It also admits to generalizations not envisioned by Langlands, for instance relating mod p Galois representations with mod p cohomology classes.

In these talks I will describe the expected Langlands correspondence in the special cases of modular curves (an example of the `self-dual' case) and arithmetic hyperbolic 3-manifolds (an example of the `non-self-dual' case). I will try both to present the general picture and to give numerical examples. I will also describe various recent theorems in the latter case due to Lan, Harris, Thorne and myself; to Peter Scholze; and to Allen, Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Thorne and myself.

Sponsored by the Rainich Lecture Series