Some results beyond the conjectures of André–Oort and Manin–Mumford

The Manin–Mumford Conjecture, a theorem of Raynaud, governs the distribution of torsion points on subvarieties of abelian varieties with respect to the Zariski topology. The conceptually related André–Oort Conjecture describes the special points on subvarieties of Shimura varieties. It was proved by Klingler, Ullmo, and Yafaev under the Generalized Riemann Hypothesis. In this talk I will describe a recent result on a combination of these two conjectures in the “mixed” setting. I will also report on joint work with Jonathan Pila on unlikely intersections beyond the André–Oort Conjecture in a product of modular curves.