Cahit Arf Lecture 2011 by Jonathan Pila

Jonathan Pila

Cahit Arf Lecture 2011

Mathematical Institute, University of Oxford

Diophantine Geometry via O-minimality

Date: November 24, 2011
at 15:40

Cahit Arf Auditorium
Department of Mathematics
Middle East Technical University
Ankara, Turkey

Supported by


The talk will be about an application of mathematical logic to certain Diophantine problems.

Diophantine geometry traditionally considers rational points on algebraic varieties. I will describe some results about the distribution of rational points on certain non-algebraic sets in real space which were developed in rough analogy with ideas in Diophantine geometry. The starting point was a result, obtained jointly with Bombieri, that rational points on the graph of a real-analytic but non-algebraic functions are "sparse" in a suitable sense. This may be generalised to a result, obtained with Wilkie, applicable to sets that are "definable in an o-minimal structure over the real field". I will define this notion, which comes from model theory, and explain why this is the natural setting. The result may be applied to certain Diophantine problems, using a strategy proposed by Zannier in the context of the Manin-Mumford conjecture. I will describe this and some further applications of this strategy to cases of the Andre-Oort and Zilber-Pink conjectures.