**Specialized course in diophantine geometry**. We shall aim at giving a detailed sketch of the proof of the Mordell conjecture following Vojta, Faltings and Bombieri. Beginning with classical major results in diophantine approximation (Thue, Roth), we shall proceed with basic material on the geometry of curves, prove an adapted version of the product theorem using a zero estimate approach and finally go to the proof of Mordell’s statement *per se*. The course will require minimal amount of prerequisites in algebraic geometry (geometry of curves and basic results on bundles over them) and height theory (Néron-Tate, functoriality), zero estimates techniques (via Bézout or arithmetic Bézout). We shall try to keep these prerequisites minimal by providing details of these prerequisites as they are needed.

*Venue*: CMI, on Friday afternoons.

**Reading groups**. We are running a reading group seminar where select papers will be chosen and studied in depth. After selecting a few papers where we believe that some deep techniques that can be of independent interest are used in an original way or that interesting research projects could be launched on connected issues with a similar approach, I will give one general lecture putting the paper in context, and suggestions of detailed division. Volunteers in the audience will then proceed in giving the subsequent lectures and upon the conclusion of the series, we shall move to the next reading.

The first paper we shall study is a result of Philip Habegger proving that the fields generated by the torsion points of an elliptic curve defined over the rationals have the Bogomolov property (*i.e. *the height is bounded below outside torsion).

*Venue*: IMSc, Wednesday afternoons.

**Upcoming**: a short series of lectures on the proof by Faltings of the Tate and Shafarevich conjectures.

*Starting date and venue* will be announced in October.