Hecke orbits on Shimura varieties of Hodge type.

Oort conjectured in 1995 that isogeny classes in the moduli space A_g of principally polarised abelian varieties in characteristic p are Zariski dense in the Newton strata containing them. There is a generalisation of this conjecture to the special fibres of Shimura varieties of Hodge type, and in this talk, I will present a proof of this conjecture under some minor assumptions. I will focus on the case of A_g since most of the new ideas can already be explained in this special case. This is joint work with Marco D'Addezio.