Time: 9h00, Wednesday, August 3, 2016
Location: Room 6, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi
Abstract: 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 and more. In 1990's Hrushovski showed how methods of model theory (a subfield of Logic)  could be applied to solve certain such problems. In 2008 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 himself 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.