O-minimality and diophantine applications


401-4037-69L - Autumn semester 2019 - Thursday 15-17 - HG G 26.1

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.


Lecture notes

The following lecture notes will be updated regularly: pdf

References
  1. 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.
  2. Jonathan Pila. O-minimality and the André-Oort conjecture for C^n. Ann. of Math. (2), 173(3):1779-1840, 2011.
  3. Jonathan Pila and Alex J. Wilkie. The rational points of a definable set. Duke Math. J., 133(3):591-616, 2006.
  4. 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.
  5. 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.
  6. Lou van den Dries. Tame Topology and O-Minimal Structures. Cambridge University Press, Cambridge, 1998.