Explore the fascinating intersection of complex dynamics and arithmetic geometry in this illuminating lecture by Prof. Laura DeMarco. Delve into the connection between dynamical systems and arithmetic geometry, starting with a simple relation between periodic points in one-dimensional systems and torsion points on two-dimensional complex tori. Discover the ideas behind DeMarco's collaborative work with Holly Krieger and Hexi Ye on the geometry of algebraic curves in genus 2. Examine elementary examples from polynomial dynamics and Julia set geometry to grasp the key components of their proofs. Learn about recent developments, including Lars Kuhne's uniform versions of Faltings' Theorem and Raynaud's Theorem, which generalize DeMarco's results. Gain insights into arithmetic equidistribution and height bounds as crucial elements in these groundbreaking mathematical advancements. This public lecture, delivered at Santa Clara University as part of the Alexanderson Award ceremony, showcases the work that earned DeMarco, Krieger, and Ye the 2020 Alexanderson Award for their paper on uniform Manin-Mumford for a family of genus 2 curves. Read more