Various extensions of Ax's functional Schanuel theorem are known, including a differential version for coverings of Shimura varieties, due to Mok-Pila-Tsimerman and one for period mappings associated with polarized variations of Hodge structures (PVHS) due to Bakker-Tsimerman. We show how these Ax-Schanuel-type theorems imply effective functional versions of Zilber-Pink-type conjectures for Shimura varieties and more generally for PVHSs. Generalizing our earlier results with Freitag on the differential equation satisfied by the j-function, we show how the differential equations associated with Shimura varieties give rise to geometrically trivial, non-aleph-nought categorical minimal types. Interestingly, the distinction between Hodge-genericity and delta-Hodge genericity is reflected in the difference between minimality and strong minimality and this puts into context our earlier example with Hrushovski

where Lascar and Morley ranks disagree.

[This is a report on on-going joint work with Jonathan Pila.]