Complex manifolds and algebraic dynamics

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Complex Manifolds and Algebraic Dynamics C. McMullen DMS-0245419 AbstractThis project investigates the behavior ofdynamical systems in settings where the methods of complex analysisand algebraic geometry can be brought to bear.The core of the proposal focuses on three types ofarea-preserving dynamical systems: * Automorphisms of algebraic surfaces, * Polygonal billiards and * Planar fluid flow.Analysis of the rich and still mysterious structure ofthese systems entails many methods, ranging from estimatesfor 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 toquantum 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 simplesystems that exhibit rich combinatorial structure and universal analytic features. Its aim is to advance the theoretical understanding of dynamics and geometry.