Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, RI
The goal of this program is to bring together researchers in complex dynamics, arithmetic dynamics, and related fields, with the purpose of stimulating interactions, promoting collaborations, making progress on fundamental problems, and developing theoretical and computational foundations on which future work will build. Complex dynamics is the study of iteration of holomorphic self-maps of a complex space. Fundamental examples of such maps arise as algebraic self-maps of algebraic varieties. Starting with the fundamental results of Fatou and Julia, complex dynamics has evolved into a well established field with many deep theorems and many important unresolved questions. Arithmetic dynamics refers to the study of number theoretic phenomena arising in dynamical systems on algebraic varieties. Many global problems in arithmetic dynamics are analogues of classical problems in the theory of Diophantine equations or arithmetic geometry, including for example uniform bounds for rational periodic points, intersections of orbits with subvarieties, height bounds and/or measure-theoretic distributions of dynamically defined sets of special points, and local-global obstructions.
While global arithmetic dynamics bears a resemblence to arithmetic geometry, the theory of p-adic (nonarchimedean) dynamics draws much of its inspiration from classical complex dynamics. As in complex dynamics, a fundamental question is to characterize orbits by their topological or metric properties. Recent progress in p-adic dynamics, especially in dimension one, has benefited from the introduction of Berkovich space into the subject.
Many computational and graphical techniques have been developed for the study of complex dynamics that have been of immense value in the development of the complex theory. Among the goals of the program will be the development of a comprehensive set of tools for studying p-adic and arithmetic dynamics.