The unlikely intersections between an arbitrary subvariety of a Shimura variety and the "special subvarieties" is expected to be governed by the Zilber-Pink conjecture. Although this conjecture remains very much open, there have been some crucial developments. Recently, work of Barroero-Dill for Shimura varieties with simple associated adjoint group, and Klingler and Tayou for more general settings, prove the conjecture when the varieties are geometrically generic. In this talk I will present joint work with V. Aslanyan and G. Fowler where we prove a somewhat similar (but different) result for the case of powers of the modular curve. Our methods differ from those of Barroero-Dill and Klingler-Tayou, instead building upon the joint work of Pila and Scanlon, which relies on the model theory of differential fields, as well as using arithmetic input in the form of Modular Mordell-Lang.