The classical Langlands program predicts that the space of automorphic functions admits a spectral decomposition along the set of Galois representations. The existence of such a decomposition was proved by V. Lafforgue. However, it does not quite tell us how to describe automorphic functions in Galois terms. In a different direction, the geometric Langlands conjecture (for D-modules) says that the category of automorphic D-modules is equivalent to (a certain modification of) the category of quasi-coherent sheaves on the space of Galois representations. It is therefore natural to try to combine these ideas to try to obtain a full description of automorphic functions using categorical methods. We will see that this indeed possible, and the key element is the assertion that the space of automorphic functions can be identified with the categorical trace of the Frobenius functor acting on the category of automorphic sheaves with nilpotent singular support.

Dennis Gaitsgory is a professor of mathematics at Harvard University. He has made important contributions to the geometric Langlands program. He is recipient of the prize of the European Mathematical Society in 2000 and the Chevalley Prize in 2018. In 2002 he was invited speaker at the International Congress of Mathematicians in Beijing. In 2020 he was elected to the National Academy of Sciences.