Rational points on Grassmannians and unlikely intersections in tori

In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. The proof uses a method introduced for the first time by Pila and Zannier to give an alternative proof of Manin-Mumford conjecture and a theorem to count points that satisfy a certain number of linear conditions with rational coefficients. This method has been largely used in many different problems in the context of "unlikely intersections".Comment: 16 page

Counting algebraic points in expansions of o-minimal structures by a dense set

The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal R}=\langle \mathcal R, P\rangle$ of $\mathcal R$ by a dense set $P$, which is either an elementary substructure of $\mathcal R$, or it is independent, as follows. If $X$ is definable in $\widetilde{\mathcal R}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is $\emptyset$-definable in $\langle \overline{\mathbb R}, P\rangle$, where $\overline {\mathbb R}$ is the real field

Model theory of special subvarieties and Schanuel-type conjectures

We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure)

Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry

Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation Theory, and Computational Geometry. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which we try to stress. Sometimes this similarity can be easily explained, sometimes the reasons remain somewhat obscure, and it motivates some natural questions discussed in the paper. We present also some new results, stressing interconnection between various types and various applications of smooth parametrization