The goal of this 3-hour tutorial is to motivate and explain in some detail the statements of various conjectures/theorems which have attracted the attention of many number theorists in recent years: from the classic conjectures of Manin-Mumford and Mordell-Lang (which became theorems in the last 30 years under the hands of Raynaud, Hindry, Faltings, McQuillan, Vojta, Hrushovski...), the more recent conjectures of Bombieri-Masser-Zannier on anomalous intersections, the conjecture of André-Oort, and their extensions by Zilber-Pink. Most of these conjectures take place in an ambient variety with a natural notion of special points and of special subvarieties and ask whether subvarieties which possess a dense subset of special points must be special. The specific notion varies among the context. The ambient variety may be an abelian variety (or more generally a commutative algebraic group), in which case special points are torsion points, or the points of a subgroup of finite rank, and special subvarieties are build up from translates of connected algebraic subgroups and ``constant'' subvarieties. The ambient variety may also be a Shimura variety, then special points and special varieties are those associated with ``maximal'' Mumford-Tate groups. The maximal generalization is due to Pink and lies within the framework of Shimura mixed varieties. The main goal of these lectures is to describe the statements of these conjectures/results. Time permitting, I will give a rough sketch of some of the proofs, so as both to make the connexion with the other lectures and to introduce the audience to the vast litterature on the topic.

The goal of this 3-hour tutorial is to motivate and explain in some detail the statements of various conjectures/theorems which have attracted the attention of many number theorists in recent years: from the classic conjectures of Manin-Mumford and Mordell-Lang (which became theorems in the last 30 years under the hands of Raynaud, Hindry, Faltings, McQuillan, Vojta, Hrushovski...), the more recent conjectures of Bombieri-Masser-Zannier on anomalous intersections, the conjecture of André-Oort, and their extensions by Zilber-Pink. Most of these conjectures take place in an ambient variety with a natural notion of special points and of special subvarieties and ask whether subvarieties which possess a dense subset of special points must be special. The specific notion varies among the context. The ambient variety may be an abelian variety (or more generally a commutative algebraic group), in which case special points are torsion points, or the points of a subgroup of finite rank, and special subvarieties are build up from translates of connected algebraic subgroups and ``constant'' subvarieties. The ambient variety may also be a Shimura variety, then special points and special varieties are those associated with ``maximal'' Mumford-Tate groups. The maximal generalization is due to Pink and lies within the framework of Shimura mixed varieties. The main goal of these lectures is to describe the statements of these conjectures/results. Time permitting, I will give a rough sketch of some of the proofs, so as both to make the connexion with the other lectures and to introduce the audience to the vast litterature on the topic.