A volcanic approach to torsion of CM elliptic curves

A celebrated theorem of Merel states that for any fixed degree $d$, there are only finitely many groups which can arise as the torsion subgroup of an elliptic curve over a number field of degree $d$. Merel's theorem followed Mazur's classification of torsion subgroups which arise over the rational numbers ($d=1$), and has been proceeded by classifications in degrees $d=2$ and $d=3$ (with a result in degree $d=4$ recently announced). I will discuss joint work with Pete Clark which allows for a complete classification in any specified degree $d$ if one restricts to torsion subgroups of elliptic curves with complex multiplication, coming from a study of isogeny volcanoes over $\overline{\mathbb{Q}}$.