**Lecturer:** Dr. Juan Diego Caycedo.

**Time and Room:** ~~Mi 8-10, SR 318~~ Mo 16-18, SR 403.

**Prerequisites:** Logic or Algebra.

**Evaluation:** There will be an oral exam at the end of the semester. Assistance to the lectures is mandatory (up to two absences allowed).

The lectures will deal with applications of the theory of o-minimalily to Diophantine geometry. More precisely, recent applications of the Pila-Wilkie theorem on counting rational points in definable sets. The basic plan is to cover the basics of o-minimality up to the cell decomposition theorem, the proof of the Pila-Wilkie theorem and, for applications, at least the proof of the Manin-Mumford conjecture (Laurent's theorem) by Pila and Zannier. Further topics may be discussed if time allows.

- O-minimality basics: from first definitions to the cell decomposition theorem. References: [5], [1], [4].
- The Pila-Wilkie theorem on counting rational points in definable sets. Reference: [2]
- The Pila-Zannier proof of the Manin-Mumford conjecture. References: [1] and [3].

- [1] D. Marker, Model Theory for Algebra and Algebraic Geometry (Lecture notes)
- [2] J. Pila and A. Wilkie, The rational points of a deļ¬nable set
- [3] J. Pila and U. Zannier, Rational points in periodic analytic sets and the Manin-Mumford conjecture
- [4] T. Scanlon, Counting special points: Logic, diophantine geometry, and transcendence theory
- [5] P. Speissegger, O-minimal structures (Lecture notes)