Many important problems in arithmetic geometry focus on the interplay between local arithmetic behavior and global geometric behavior. A way of making this precise is by studying so-called unlikely intersections, which are presumably governed by the (very open) Zilber-Pink conjecture.
In this talk I will introduce the Zilber-Pink conjecture, discuss some cases that are known, and present joint work with Thomas Scanlon where we prove a strong counterpart of Zilber-Pink describing the presence of likely intersections.