COMS

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.