Many geometric spaces carry natural collections of special submanifolds that encode their internal symmetries. Examples include abelian varieties and their sub-abelian varieties, locally symmetric spaces with their totally geodesic subspaces, period domains with their sub–period domains, and strata of abelian differentials with their affine invariant submanifolds.

In recent years, major progress has been made in understanding these structures through the framework of unlikely intersections and functional transcendence in foliated bundles. I will survey how this perspective can be applied to the study of non-arithmetic complex hyperbolic lattices and affine invariant submanifolds, complementing existing dynamical approaches.