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 Zannier 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.

The following lecture notes will be updated regularly: pdf

- Gareth O. Jones and Alex J. Wilkie, editors.
*O-Minimality and Diophantine Geometry*, volume 421 of*London Mathematical Society Lecture Note Series*. Cambridge University Press, Cambridge, 2015. - Jonathan Pila. O-minimality and the André-Oort conjecture for C^n.
*Ann. of Math*. (2), 173(3):1779-1840, 2011. - Jonathan Pila and Alex J. Wilkie. The rational points of a definable set.
*Duke Math. J.*, 133(3):591-616, 2006. - Jonathan Pila and Umberto Zannier. Rational points in periodic analytic sets and the Manin-Mumford conjecture.
*Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.*, 19(2):149-162, 2008. - Thomas Scanlon. O-minimality as an approach to the André-Oort conjecture. In
*Around the Zilber-Pink Conjecture/Autour de La Conjecture de Zilber-Pink*, volume 52 of*Panor. Synthèses*, pages 111-165. Soc. Math. France, Paris, 2017. - Lou van den Dries.
*Tame Topology and O-Minimal Structures*. Cambridge University Press, Cambridge, 1998.