Complex manifolds and algebraic dynamics

Harvard University, Cambridge MA

Investigators

Abstract

Complex Manifolds and Algebraic Dynamics C. McMullen DMS-0245419 Abstract This project investigates the behavior of dynamical systems in settings where the methods of complex analysis and algebraic geometry can be brought to bear. The core of the proposal focuses on three types of area-preserving dynamical systems: * Automorphisms of algebraic surfaces, * Polygonal billiards and * Planar fluid flow. Analysis of the rich and still mysterious structure of these systems entails many methods, ranging from estimates for differential equations with minimal smoothness to rigidity results for Lie groups acting on bundles over the moduli space of Riemann surfaces. From meteorology to robotics, from fluid mechanics to quantum field theory, the control and prediction of dynamical systems plays a central role in science and technology. This project will include computer explorations of mathematically simple systems that exhibit rich combinatorial structure and universal analytic features. Its aim is to advance the theoretical understanding of dynamics and geometry.