In the past few years, conjectures have been made and partial results achieved on unlikely intersections in complex dynamics, following the program initiated by Baker-DeMarco. I will explain their dynamical generalization of the famous André-Oort conjecture on CM points in moduli spaces of abelian varieties. Ghioca, Nguyen, Ye, and myself recently proved the first complete case of this conjecture, for pairs of unicritical polynomials, and I will discuss our result and the connection to invariant subvarieties of $P1 \times P1$, and the structure of the Mandelbrot set.[-]

We will present work in progress, joint with Hexi Ye, towards a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds for common torsion points of nonisomorphic elliptic curves. We introduce a general approach towards uniform unlikely intersection bounds based on an adelic height pairing, and discuss the utilization of this approach for uniform bounds on common preperiodic points of dynamical systems, including torsion points of elliptic curves.[-]