Abstract
The results outlined in this talk are part of a wider, flourishing
interaction between diophantine geometry and model theory. The
central aim is to bound the density of rational and algebraic points
lying on certain `transcendental' subsets of the reals, with a view
to diophantine applications. Following influential work by Pila and
Wilkie in this area (which has had stunning number-theoretic
applications e.g. to the Manin-Mumford and André-Oort Conjectures),
our focus is on sets which are first-order definable in various
`o-minimal' expansions of the real field. We shall survey background
and some results in this area, in particular concerning possible
improvements to the bound given by Pila and Wilkie. These include
instances of a conjecture of Wilkie, which proposes an improvement
for the real exponential field, and some recent progress made
towards finding an effective version of the Pila-Wilkie Theorem.