Title: Uniform Boundedness Conjectures and Theorems for Dynamical Systems

Abstract: The standard uniform boundedness conjecture for preperiodic points of dynamical systems is, in some sense, a dynamical analog of Mazur's theorem that elliptic curves over Q have bounded torsion and Merel's extension of that result to number fields of fixed degree. However, the tools used in proving Mazur's and Merel's theorems are not applicable in the case of dynamical systems.

Changing the problem slightly, it turns out, enables us to prove much stronger results. If we concern ourselves with precritical points rather than preperiodic points, and if we restrict our attention to the case of quadratic polynomials, we can prove a rather strong uniform boundedness theorem.

This talk will describe the bigger Uniform Boundedness Conjecture for preperiodic points, as well as describe recent results regarding uniform boundedness of precritical points. I will not assume any background in dynamics. This is joint work with several other researchers, coming out of a working group at a recent AIM workshop.