Summer School on the Langlands Program

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This award will fund US-based graduate students and postdocs to participate at the Summer School on the Langlands program, to be held in July 2022 at the Institut des hautes études scientifiques (IHES) in France. The Langlands program is one of the most expansive and consequential research programs in today's mathematics, spanning a breathtaking range of topics, from number theory to mathematical physics, and the summer school aims to highlight connections between various subfields, and train future researchers in this demanding and multifaceted area. At its heart, the Langlands program aims to understand solutions of diophantine equations (polynomial equations with integer coefficients) by establishing connections with the frequencies of certain higher-dimensional "drums" called "arithmetic manifolds." The modularity conjecture, which led to the proof of Fermat's "last theorem," is one of the instances of such connections. Many of the world's foremost experts on the subject will be speaking in the summer school; the event is taking place 45 years after a historic summer school in Corvallis, Oregon, where the experts of the time set the agenda of the Langlands program for several decades. Meeting website: https://indico.math.cnrs.fr/event/6909/Although the Langlands program has expanded too much to be covered in its entirety, the summer school will highlight connections between the representation-theoretic, arithmetic, and geometric aspects of it. On the analytic side, there will be updates on the status of endoscopy and beyond, trace formulas, the theory of periods of automorphic forms and L-functions, and explicit methods for proving functoriality. On the geometric side, developments around local and global Shimura varieties and shtukas will be covered, together with recent breakthroughs on the Langlands conjectures over function fields, the local Langlands conjectures, and relations to the geometric Langlands program. On the arithmetic side, new geometric insights on moduli spaces of Langlands parameters and Galois representations will be discussed, together with their connections to modularity lifting theorems and derived structuresThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.