Title: (Un)likely intersections

Abstract: The Zilber-Pink conjectures predicts that for an ambient special variety  (such as an abelian variety or a Shimura variety), if   is an irreducible algebraic subvariety which is not contained a proper special subvariety of  (e.g. a proper algebraic subgroup in the abelian variety case or a variety of Hodge type in the case of Shimura varieties), then the union of the unlikely intersections  as  ranges over the special subvarieties of  with  is not Zariski dense in .  While various instances of this conjecture have been proven, it remains open in most cases of interest.  In this lecture, I will describe some of my work with Jonathan Pila in which we prove an effective function field version of this conjecture along with a counterpart to the Zilber-Pink conjecture proven with Sebastian Eterović:  after accounting for some geometric obstructions, the likely intersections, i.e. the union of the intersections  with  special and ,  are dense in the Euclidean topology in .   Our techniques for both results come from o-minimal complex analysis and differential algebra.