Introduction to Nevanlinna Theory,
Hyperbolicity and Arithmetic Dynamic

In the 1980s, building on earlier insights by Osgood and Lang, Vojta developed a "dictionary" that bridges Diophantine approximation with Nevanlinna theory. For example, the existence of infinitely many integral or rational points on a variety \( X \) corresponds to the existence of a non-constant holomorphic map \( f : \mathbb{C} \to X \). These analogies form the basis for profound connections between arithmetic and complex geometry.
This mini-course will introduce key topics in Nevanlinna theory and complex hyperbolicity, laying the groundwork for understanding value distribution and detecting algebraic degeneracy of entire curves in both complex analysis and complex geometry. It will also cover the fundamentals of (function field) algebraic hyperbolicity, which sits at the intersection of algebraic geometry, Diophantine geometry, and complex geometry. Finally, the course will offer an introduction to arithmetic dynamics, an exciting area of modern research that combines number theory and dynamical systems, fostering rich interactions between complex dynamics, Diophantine geometry, and Diophantine approximation.

Liang-Chung Hsia (National Taiwan Normal University)
Erwan Rousseau (Brest University)
Min Ru (University of Houston)
Amos Turchet (Roma Tre University)