The periodic orbits and their structure are fundamental features of a dynamical system. In an algebraic setting, where the system is defined by polynomials, we can use tools from algebraic or arithmetic geometry to study these orbits. Important special cases include endomorphisms of abelian varieties and play a role in some exciting developments in arithmetic geometry (for example in the uniform versions of the Mordell or Manin-Mumford Conjectures in the recent work of Dimitrov-Gao-Habegger, Kühne, Yuan and others), where the torsion points of the group coincide with the preperiodic points of an endomorphism. In this talk, I will describe some parallel questions and recent progress on more general families of complex and arithmetic dynamical systems.

Laura DeMarco is a Professor of Mathematics at Harvard University. She is a Fellow of the American Mathematical Society, a Ruth Lyttle Satter prize winner, a member of the National Academy of Sciences, an ICM speaker, and together with Holly Krieger and Hexi Yu, received the 2020 Alexanderson Award.