Résume | The André-Oort conjecture is an important problem in arithmetic geometry concerning subvarieties of Shimura varieties. It attempts to characterise those subvarieties $V$ for which the special points lying on $V$ constitute a Zariski dense subset. When the ambient Shimura variety is the moduli space of principally polarised abelian varieties of dimension $g$ (in which case, a special point is a point corresponding to the isomorphism class of a CM abelian variety), the conjecture has been obtained by Pila and Tsimerman via the so-called Pila-Zannier strategy, reliant on the Pila-Wilkie counting theorem on o-minimal structures. In this talk, we will outline the Pila-Zannier strategy, providing some introduction to Shimura varieties and André-Oort, and explain the state of the art for the full conjecture. In particular, we will mention certain height bounds obtained jointly with Orr. |