In this talk, we will be working in with the theory of differentially closed fields of characteristic zero; essentially this theory says that every differential equation which might have a solution in some field extension already has a solution in the differentially closed field. After introducing this theory in a bit of detail, we will sketch a proof of the strong minimality of the differential equation satisfied by the classical $j$-function starting from Pila's modular Ax-Lindemann-Weierstrass theorem. This resolves an open question about the existence of a geometrically trivial strongly minimal set which is not $\aleph _0$-categorical. If time allows, we will discuss some finiteness applications for intersections of certain sets in modular curves. This is joint work with Tom Scanlon.