In this talk, I will present a connection between dynamical systems and arithmetic geometry. Building on a simple relation -- between periodic points for a particular class of systems in dimension 1 and the torsion points for the group structure on a 2-dimensional complex torus -- I will describe some ideas that went into my joint work with Holly Krieger and Hexi Ye about the geometry of algebraic curves in genus 2. Most of the talk will be devoted to a more elementary setting, coming from the dynamics of polynomials and the geometry of Julia sets, to illustrate the ingredients in our proofs. In an exciting development since our work appeared, Lars Kuhne was able to prove uniform versions of Faltings' Theorem (the Mordell Conjecture) and of Raynaud's Theorem (the Manin-Mumford Conjecture) this year, vastly generalizing our results; some of the ingredients in his proofs (arithmetic equidistribution and height bounds) are similar, and I will emphasize some of these aspects.

**Note**:

This is a video of the public lecture delivered by Prof. Laura DeMarco at Santa Clara University as a part of the Alexanderson Award ceremony on September 30, 2021.

Laura DeMarco, Holly Krieger, and Hexi Ye were the recipients of the 2020 Alexanderson Award (awarded by The American Institute of Mathematics) for their paper, “Uniform Manin-Mumford for a family of genus 2 curves,” which was published in the Annals of Mathematics in 2020. The paper grew out of the AIM SQuaRE, Dynamical Andre-Oort Questions.

This colloquium is part of the Distinguished Lecture Series (DLS) organized by The (Indian) Mathematics Consortium (TMC), and co-hosted by IIT-B and ICTS-TIFR. For more information about this lecture series, and to register, please visit the homepage of the TMC DLS.