Height pairings and preperiodic points in P^1

In 2011, Matt Baker and I used height functions and arithmetic equidistribution to show when two complex rational functions have only finitely many (pre-)periodic points in common.  In this talk, I will explain how to obtain uniform bounds in families, inspired by questions and conjectures about elliptic curves and their torsion points.  The key ingredient is an estimate on the Arakelov-Zhang intersection numbers in families of heights.  This is joint work with Holly Krieger and Hexi Ye.