Canonical heights and the Andre-Oort conjecture

Let S be a Shimura variety. The Andre-Oort conjecture posits that the Zariski closure of special points must be a sub Shimura subvariety of S. The Andre-Oort conjecture for A_g (the moduli space of principally polarized Abelian varieties) — and therefore its sub Shimura varieties — was proved by Jacob Tsimerman. However, this conjecture was unknown for Shimura varieties without a moduli interpretation. I will describe joint work with Jonathan Pila and Jacob Tsimerman (with an appendix by Esnault-Groechenig) where we prove the Andre Oort conjecture in full generality.