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Let G be a semisimple algebraic group defined over the field Q of rational numbers and let G(Q) denote
the group of rational points of G. Then G(Q) can be regarded as a discrete subgroup of the locally compact
group G(A) of adelic points of G. Moreover, the group G(A) carries a canonical (bi-invariant) measure,
called Tamagawa measure. A celebrated conjecture of Weil asserts that, if the group G is simply connected,
then the Tamagawa measure of the quotient G(Q)\G(A) is equal to 1. Weil’s conjecture is now a theorem
of Kottwitz, building on earlier work of Langlands and Lai. More recently, Gaitsgory and Lurie proved a
version of Weil’s conjecture in the setting of function fields, using techniques inspired by algebraic topology
(specifically, the theory of factorization homology). The goal of this lecture series is to explain some of the
ideas surrounding the proof.


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