ABSTRACT::

In part I of the mini-course we will present the model theoretic proof of Gabrielov theorem of the complement in real sub analytic geometry. This model theoretic proof (due to Denef and van den Dries) is based on quantifier elimination (QE) method is shorter and more explicit then the original proof, it used Weierstrass preparation theorem and Tarski-Seidenberg QE for the real field.  If time allows we will also present a (more geometric) proof  of Tarski QE result.



In part II we will present the model theoretic proof (again due to Denef and van den Dries) of Hironaka's p-adic analogue of Gabrielov theorem. As before this other method is shorter, more explicit and based on Weierstrass preparation and Macintyre QE for the p-adic field.



In part III we will talk about the number theoretic applications of the above,  respectively: (i) the recent  new proof of Manin-Mumford (and in fact, extending the method, of Andre-Ort type conjectures) (Pila-Zanier, Pila-Wilkie, Peterzil-Starchenko); (ii) the old proof of rationality of the Poincaré series of closed analytic/algebraic sets of p-adic integers.



We hope to present the subject in a way accessible to everyone (including the speaker!), but with details and small exercises for the basic steps. Details and small exercises will probably not show up in part III.