Riemann Surfaces, Dynamics and Hyperbolic Geometry

黎曼曲面、动力学和双曲几何。麦克马伦DMS-9806424这个项目将研究几何结构和动力系统的分类，在允许对具体例子进行协调研究的环境中。它的范围包括迭代有理映射、Klein群、黎曼曲面上的射影结构和射影平面上的代数动力学。重整化，谱理论与复动力学的相互作用，以及TeichMuller理论在三维流形上的应用是主要的研究方向。从气象学到机器人学，动力系统的控制和预测在科学技术中发挥着核心作用。这个项目将包括对数学上简单的系统的计算机探索，这些系统展示了丰富的组合结构和普遍的分析特征。其目的是促进对动力学和几何学的理论理解。

Riemann Surfaces, Dynamics and Hyperbolic Geometry-C. McMullen DMS-9806424 This project will study the classification of geometric structures and dynamical systems, in settings that allow for a concerted study of specific examples. Its scope includes iterated rational maps, Kleinian groups, projective structures on Riemann surfaces, and algebraic dynamics on the projective plane. Renormalization, the interaction of spectral theory with complex dynamics, and applications of Teichmuller theory to 3-manifolds are among the main research directions. From meteorology to robotics, 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.