The Hecke orbit conjecture

I will talk about the proof of the Hecke orbit conjecture for Shimura varieties in characteristic p, as formulated by Chai and Oort. After reviewing the statement in the case of the moduli space of abelian varieties, I will explain the linearisation of the problem using generalised Serre-Tate coordinates. I will then present the role of monodromy groups of isocrystals in this context. In the end, I will explain how the conjecture gives the surjectivity of the period map for K3 surfaces. This is a joint work with Pol van Hoften.