Complex manifolds and algebraic dynamics

复杂流形和代数动力学本项目研究在复杂分析和代数几何方法可以发挥作用的情况下动力系统的行为。该建议的核心集中在三种类型的保面积动力系统：代数表面的自同构，多边形台球和平面流体流动。分析这些系统丰富而神秘的结构需要许多方法，从最小光滑微分方程的估计到黎曼曲面模空间上作用于束的李群的刚性结果。从气象学到机器人，从流体力学到量子场论，动力系统的控制和预测在科学技术中起着核心作用。这个项目将包括对数学简单系统的计算机探索，这些系统表现出丰富的组合结构和普遍的分析特征。它的目的是促进动力学和几何学的理论认识。

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.