Event № 293
I will describe a long line of research around the asymptotic density of rational points in transcendental varieties, starting from the work of Bombieri-Pila on analytic curves in the late eighties and on to its vast generalization in the work of Pila-Wilkie about ten years ago. The latter sits at the crossroads between analysis, logic and diophantine geometry, and has attracted considerable attention in the last decade after playing the key role in new proofs of several conjectures on unlikely arithmetic intersections, including the first proof of the Andre-Oort conjecture (by Pila). I will give a taste of the philosophy of these applications in an elementary example. Finally I will discuss one of the main open problems of the area, the Wilkie conjecture, and describe some recent progress obtained in a joint work with Dmitry Novikov.