Title: Additive combinatorics and descent
Abstract: In this talk I will outline a method introduced in joint work with Peter Koymans that allowed us to settle Hilbert 10th problem for all finitely generated rings and to show that every number field has an elliptic curve of rank 1. I will also present joint work with Alexandra Shlapentokh outlining some further consequences of these results and showing, among other things, that one can define the rationals over any number field with a formula employing exactly one universal quantifier.