We describe a recent program for analyzing definable sets and groups in certain model theoretic settings. Those settings include:(a) o-minimal structures (M, P), where M is an ordered group and P is a real closed field defined on a bounded interval (joint work with Peterzil),(b) tame expansions (M, P) of a real closed field M by a predicate P, such as expansions with o-minimal open core (work in progress with Gunaydin and Hieronymi).The analysis of definable groups first goes through a local level, where a pertinent notion of a pregeometry […]

The Tate-Voloch conjecture is a statement about p-adic distance from torsion points to subvarieties in a semi-abelian variety defined over C_p. The use of Galois equations on torsion points by Pink and Rossler to prove the Manin-Mumford conjecture can be adapted to prove that conjecture in the case where both the semi-abelian variety and its subvariety are defined over a finite extension of Q_p.In this talk, we will present such a proof, and try to give an insight on how this proof differs from the model-theoretic one given by Scanlon.

In the spirit of work by Pila-Wilkie (2006) and by Pila (2009), we will present bounds on the number of points of bounded height in the non-archimedean context. An important tool to make the determinant method work is provided by a non-archimedean version of the Yomdin - Gromov parameterizing lemma. We wil explain these results, obtained in joint work with G. Comte and F. Loeser.