Number/Representation Theory

Inspired by the analogy between torsion points on abelian varieties and preperiodic points of rational maps, Shou-Wu Zhang formulated a dynamical generalization of the Manin–Mumford and Bogomolov conjectures. These conjectures aim to classify subvarieties of $\mathbb{P}^n$ that contain a Zariski-dense or “generic” sequence of preperiodic points for an endomorphism $f$ of $\mathbb{P}^n$ – or more generally, points of small canonical height with respect to $f$. Despite significant progress in special cases, the conjectures remain largely open. In this talk, we will discuss recent progress towards uniform and quantitative versions of the dynamical Bogomolov conjecture. This is joint work in progress with Jit Wu Yap.