Maximal compact subvarieties of ${\mathcal A}_g$

We present results on the maximal dimension of compact subvarieties of the moduli space of abelian varieties and of moduli of complex curves of compact type. Equivalently, this is the maximal dimension of a compact complex parameter space for a maximally varying family of abelian varieties/curves, etc. Based on joint work with Mondello, Salvati Manni, Tsimerman.