Gareth Jones, of the University of Manchester, presents a research talk from the IAS workshop on Recent Developments in Hodge Theory and O-minimality. He opens with joint work with Binyamini, Schmidt, and Thomas proving a uniform Manin-Mumford result for products of CM elliptic curves, a statement about how torsion points can distribute across such products. Jones then shows how this feeds into an effective Andre-Oort result for fibre powers of the Legendre family, a problem concerning special points on families of elliptic curves and the bounds that can be put on them. He closes with work in progress alongside Schmidt extending these methods to multiplicative extensions. The talk assumes familiarity with o-minimality, Hodge theory, and arithmetic geometry, and is aimed at specialists following the current push toward effective versions of classical special point conjectures. Delivered at Simonyi Hall 101, it is a technical seminar presentation rather than an introductory lecture, tracking live research rather than summarizing settled results.