Le lezioni si terranno il mercoledì dalle 16 alle 18 e il venerdì dalle 11 alle 13, in Saletta Riunioni.
The application of model-theoretic methods to theories of fields has provided numerous insights over the years, from Ax’s proof that injective polynomial maps from complex affine space to itself are also surjective to the recent applications of o-minimality to algebraic geometry and number theory. The aim of this course is to introduce various model-theoretic tools and notions through their application to first-order theories of fields. We will focus on the theories of algebraically closed fields, real closed fields, differential fields, and exponential fields, showing what model-theoretic methods can tell us about these structures and taking the opportunity to introduce the basics of classification theory. More concretely, we will see statements of the form: “definable subsets of an algebraically closed field are finite or have finite complement” and “definable subsets of a real closed field are finite unions of points and intervals” and see what their consequences are. Depending on the interest of the audience, towards the end we will cover more advanced topics, possibly Zilber’s conjecture on definability in the complex exponential field or the applications of the Pila-Wilkie theorem in Diophantine geometry.