Title: Anabelian geometry via étale sites

Abstract: Given a variety X over a field k, one can associate to X its étale fundamental group \pi_1(X), an algebro-geometric analogue of the topological fundamental group. Grothendieck conjectured that if X is defined over \mathbb{Q} and \pi_1(X) is sufficiently “non-abelian”, then X should be reconstructible from \pi_1(X). This applies to hyperbolic curves—those of genus at least two—but higher-dimensional varieties for which this holds have been difficult to find.

In his Esquisse d’un programme and his letter to Faltings, Grothendieck proposed a broader anabelian conjecture: that any variety over a field finitely generated over \mathbb{Q} should be reconstructible from its étale site, a generalization of the étale fundamental group.

In this talk, I will give an accessible introduction to anabelian geometry. I will then sketch a proof of Grothendieck’s generalized conjecture, joint with Peter Haine and Sebastian Wolf, followed by an extension which is joint with Jakob Stix. I will conclude with new results that highlight the strength of the étale site as an algebro-geometric invariant.