<p><span style="color: rgb(36, 36, 36);">In the 1980s, Raynaud proved a theorem (the Manin-Mumford Conjecture) about the geometry of torsion points in abelian varieties, with number-theoretic methods. Around the same time, and with completely different methods, McMullen proved an important dynamical rigidity theorem for holomorphic maps on P^1. In joint work with Myrto Mavraki, we explained how to view these results as special cases of a unifying conjecture. (Our conjectural statement was inspired by a recent theorem of Gao and Habegger, called "Relative Manin-Mumford", and results in complex dynamics of Dujardin, Gauthier, Vigny, and others.) I will present the conjecture and what is known, with as many examples as I have time for.</span></p>