Recent progress on Langlands reciprocity for $\mathrm{GL}_n$: Shimura varieties and beyond

Abstract

The goal of these lecture notes is to survey progress on the global Langlands reciprocity conjecture for $\mathrm{GL}_n$ over number fields from the last decade and a half. We highlight results and conjectures on Shimura varieties and more general locally symmetric spaces, with a view towards the Calegari-Geraghty method to prove modularity lifting theorems beyond the classical setting of Taylor-Wiles.