Yesterday I was at Peter Scholze’s Seminaire Bourbaki talk, his write-up is available here. It was advertised as an exposition of the recent proof of the geometric Langlands conjecture, but Pierre Colmez accurately describes it as something much more: the Langlands program 2.0.
In his talk, Scholze described the situation of the geometric Langlands program as for many years getting further and further away from the original arithmetic Langlands program. Recently there has been a dramatic change, as things have turned around and geometric Langlands is having an impact on arithmetic Langlands. He gives a new version of the basic conjecture of the arithmetic Langlands program (Conjecture 1.5), in a form similar to that of geometric Langlands. He describes this conjecture though as merely a “template for a conjecture”, since some of the terms remain undefined.
I learned from his references about something perhaps more accessible, David Ben-Zvi’s talk What is the Geometric Langlands Correspondence About? at the AMS meeting this past January. The blurb for the talk is:
Number theorists found the tusks. Physicists found the tail. Now geometric Langlands tells us it’s an elephant.
