Appears in collection : Rational points on irrational varieties

I will talk about some results old and new about the relationship between rational points on varieties and fundamental groups. In a paper with Hast, we extend the class of curves to which Kim’s non-abelian Chabauty method can be applied by means of a strange and interesting result of Bogomolov and Tschinkel about covers of hyperelliptic curves. I will also talk about old unpublished work with Venkatesh about uniform bounds for points of bounded height on varieties with large fundamental groups, in the hopes that people in the audience will have some ideas for contexts in which such a result might be useful!