Clay Mathematics Institute

Dedicated to increasing and disseminating mathematical knowledge

On a Bombieri-Masser-Zannier theorem

Laura Capuano (Scuola Normale Superiore)

Abstract: In this seminar we present an alternative proof of a theorem of Bombieri, Masser and Zannier of 1999 about intersecting a curve in the multiplicative group $\mathbb{G}_m^n$ with all the algebraic subgroups of of dimension n-2.  To do that, we use a method introduced for the first time in 2008 by Pila and Zannier to give a new proof of Manin-Mumford conjecture and a theorem of Pila about counting rational points on certain transcendental varieties of Grassmannians. This method has been used afterwards to prove other cases of Unlikely Intersections problems in many different contexts.