Departmental Colloquium

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

Post Critically finite maps (PCF) are those whose critical points are preperiodic -- they play a special role within the moduli space \mathcal{M}_d of degree d rational maps. In this talk we will discuss the dynamical Andr\'e-Oort Conjecture (DAO), which asks for a classification of the PCF-special subvarieties in \mathcal{M}_d. DAO was recently proven in the case of curves by Ji-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'.