Seminar

The results outlined in this talk are part of a wider, flourishing interaction between diophantine geometry and model theory, in particular o-minimality. We are concerned with bounds on the density of rational and algebraic points lying on certain subsets of the reals - more specifically those which are first-order definable in various o-minimal expansions of the real field. Following influential work by Pila and Wilkie in this area, Wilkie conjectured an improvement to their main result for sets definable in the (o-minimal) real exponential field. This conjecture we established in the case of curves and certain surfaces (joint with Jones). We will briefly review the context and then outline these results; time permitting we will also consider some related results and applications.