The Pila-Wilkie Theorem and variations
Introductory Workshop: Model Theory, Arithmetic Geometry and Number Theory February 03, 2014 - February 07, 2014
Location: MSRI: Simons Auditorium
Pila and Wilkie's influential counting theorem provides a bound on the density of rational points of bounded height lying on the 'transcendental parts' of sets definable in o-minimal expansions of the real field. This result has brought about a lively interaction in recent years between o-minimality and diophantine geometry, including several important applications to arithmetical conjectures which will be explored further in Ya'acov Peterzil's tutorial. As a prelude to this, we will provide an introduction to the Pila-Wilkie Theorem, indicating the main ingredients involved in the proof. In particular, we will focus on the key step known as the Pila-Wilkie Reparameterization Theorem. This is a model theoretic statement about the geometry of sets definable in o-minimal expansions of real closed fields - namely that they can be covered by the images of finitely many sufficiently differentiable functions with bounded derivatives. Following the Pila-Wilkie Theorem, subsequent work carried out by a number of authors, including Pila, Besson, Boxall, Butler, Jones, Masser and myself, has focussed on establishing that a sharper bound holds in certain situations. One important goal is a conjecture of Wilkie concerning sets definable in the real exponential field. We shall explore some of the cases of this conjecture already established and the methods involved, indicating how a suitable modification of the Pila-Wilkie notion of parameterization could play an important role in the pursuit of this conjecture.
If none of the options work for you, you can always buy the DVD of this lecture. The videos are sold at cost for $20USD (shipping included). Please Click Here to send an email to MSRI to purchase the DVD.
See more of our Streaming videos on our main VMath Videos page.