A family of problems in Diophantine geometry has the following

form: We fix a collection of "special" algebraic varieties among which the

0-dimensional are called "special points". Mostly, if V is a special variety

then the special points are Zariski dense in V, and the problem is to prove

the converse: If V is an irreducible algebraic variety and the special

points are Zariski dense in V then V itself is special.

Particular cases of the above are the Manin-Mumford conjecture

(where the special varieties are certain cosets of abelian sub-varieties and

the special points are torsion points), the Mordell-Lang conjecture, the Andre-Oort Conjecture. In 1990's Hrushovski showed how methods of model theory could be applied to solve certain such problems. About 10 years ago Pila and Zannier developed a different framework which allows to apply model theory and especially the theory of o-minimal structures, to tackle questions of this nature over the complex numbers. Pila used these methods to prove some open cases of the Andre-Oort conjecture and since then there was an influx of articles which use similar techniques. At the heart

of the Pila-Zannier method lies a theorem of Pila and Wilkie on rational

points on so-called definable sets in o-minimal structures.

In these survey talks I will describe the basic mode theoretic ingredients of

the Pila-Zannier method and the way in which it is applied. I

will not assume prior knowledge of model theory or Logic and will try to

explain all notions which may come up.

Zoom link:

https://huji.zoom.us/j/87131022302?pwd=SnRwSFRDNXZ4QVFKSnJ1Wit2cjZtdz09