11/11/2009
Jonathan Pila
University of Bristol
A model-theoretic approach to certain diophantine problems
I will describe some results and problems about the distribution of rational points on certain non-algebraic sets in real space. They find their natural setting in the model-theoretic notion of an `o-minimal structure over the real numbers'. I will describe a result joint with Alex Wilkie in this setting. A surprising strategy, proposed by Umberto Zannier, uses this result to approach diophantine problems in the Manin-Mumford-Andre-Oort circle of conjectures. I will describe some implementations of this strategy highlighting the use of o-minimality at different junctures, and discuss its prospects.