Many problems in arithmetic geometry have the following form: given a subvariety X of a variety M and a subset S of X, can one describe the structure of the components of the Zariski closure of S? These questions become particularly interesting when the set  has some `special' structure (perhaps related to a group law on M). The expectation is then that the components of the closure of S will inherit this structure and be `special' themselves. Examples of problems in this form, called `unlikely intersections', include the Manin-Mumford conjecture, the Mordell-Lang  conjecture and the Andre-Oort conjecture.

Post-critically finite maps  (PCF) are those whose critical points are preperiodic -- they play a special role within the moduli space  of degree d rational maps. In this talk we will discuss the dynamical Andre-Oort Conjecture (DAO), which asks for a classification of PCF-special subvarieties. DAO was recently proven in the case of curves by Ji and Xie, following works by many  authors, but remains open in higher dimensions. We will discuss results obtained with L.  DeMarco and H. Ye, on bounding the geometry of the PCF-special subvarieties. Our results can be thought of as a `uniform DAO'.

Speaker: Myrto Mavraki, University of Toronto