O-minimal structures provide a framework for tame topology as envisioned by Grothendieck. Originally it was mainly a topic of interest for real algebraic geometers. However, since Pila and Wilkie proved their counting theorem for rational points of bounded height, many applications to diophantine and algebraic geometry have been found.
The overall goal of this course is to provide an introduction to o-minimality and to prove results needed for diophantine applications.
The first part of the course will be devoted to the definition of o-minimal structures and to prove the cell decomposition theorem, which is crucial for describing the shape of subsets of an o-minimal structure. In the second part of the course, we will prove the Pila-Wilkie counting theorem. The last part will be devoted to diophantine applications, with the proof by Pila and Zanier of the Manin-Mumford conjecture and, if time permit, a sketch of the proof by Pila of the André-Oort conjecture for product of modular curves.
G. Jones and A. Wilkie: O-minimality and diophantine geometry, Cambridge University Press L. van den Dries: Tame topology and o-minimal structures, Cambridge University Press
Prerequisites / Notice
This course is appropriate for people with basic knowledge of commutative algebra and algebraic geometry. Knowledge of mathematical logic is welcomed but not required.