The Zilber-Pink conjectures encompass a whole array of problems, originating in the Mordell conjecture, and including the conjectures of Manin-Mumford and Mordell-Lang, the André-Oort conjecture, and questions on unlikely intersections raised by Bombieri-Masser-Zannier and by Zilber. Given a semi-abelian scheme or more generally, a mixed Shimura variety S, one studies how an algebraic subvariety intersects the “special” subvarieties of S. Following the standard motto of diophantine geometry, the conjectures express that the geometry of S should govern its arithmetic. For instance, if a curve meets the special subvarieties of codimension at least 2 infinitely often, then it should lie in a strict special subvariety of S.

Important progress has been made on these problems in the last decade, based on a variety of approaches which has turned the Zilber-Pink conjectures into a meeting point of classical diophantine methods (heights, Lehmer-type problems), arithmetic geometry (density of Galois orbits), Hodge theory (Mumford-Tate and monodromy groups), and model theory (o-minimality).

The objective of the summer school is two-fold. On the one hand, three courses will be given during the first week, providing the audience with an introduction to the theories of Shimura varieties, of heights on algebraic groups, and of o-minimality. On the other hand, shorter courses, lectures and problem sessions, will be held during both weeks, giving the state of the art on the conjectures, and listing open (and hopefully accessible) research problems connected to these topics.

In addition, a satellite day will take place on Saturday 30 June 2012.