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'.