Title: Smooth parameterization in o-minimal structures
Speaker: Margaret Thomas (University of Konstanz)
Abstract: The counting theorem of Pila and Wilkie opened up one of the most important developments in model theory in recent years. It provides a bound on the density of rational points for sets definable in o-minimal expansions of the real field, a result which has had several stunning number-theoretic applications (e.g. to the Manin-Mumford and André-Oort Conjectures). Central to the proof of the theorem is an o-minimal version of Yomdin-Gromov parameterization, a type of `smooth parameterization', the decomposition of sets using functions with controlled higher-order derivatives. We will discuss various different directions of current research relating smooth parameterization and o-minimality. One direction is the pursuit of an effective version of the Pila-Wilkie Counting Theorem, and our first results in this direction are for certain surfaces described by so-called Pfaffian functions. Another direction is the study of `mild parameterization' in o-minimal structures. This is aimed towards a conjecture of Wilkie, which proposes a significant sharpening of the Pila-Wilkie bound for sets definable in the (o-minimal) real exponential field, and which would be established with sufficiently uniform mild parameterization in this setting.