Basic Notions: Kobi Peterzil (U. of Haifa) "The Pila-Zannier method: applications of model theory to Diophantine geometry".

 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:


Thu, 07/01/2021 - 16:00 to 17:15