The Langlands program and modern geometry

The Langlands program predicts some surprising connections between number theory and representation theory, most of which are still conjectural. In recent years, powerful new tools in non-archimedean geometry have led to dramatic progress on these questions. One fascinating aspect of these new tools is that in a certain precise sense, they allow us to treat a fixed prime number p as a variable, which opens the door to importing ideas from the geometric Langlands program into more arithmetic settings. I will give a historically motivated introduction to this circle of ideas, and give a big-picture view of some of the most exciting recent and ongoing developments.