Complex dynamics and moduli spaces

复杂动力学和模空间

• 批准号: 0755765
• 负责人: Curtis McMullen
• 金额: $ 93.79万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0755765&HistoricalAwards=false
• 关键词: Complex; dynamics; moduli; spaces

This project concerns the interplay between dynamical systems, complex analysis, and the algebra, geometry and topology of moduli spaces. The settings for this investigation include(i) The moduli space of Riemann surfaces M_g, its complex geodesics, and the bundle of holomorphic 1-forms;(ii) Billiards in rational polygons;(iii) Real and complex K3 surfaces, their automorphisms and their moduli as encoded by Hodge structures;(iv) Automorphisms of rational surfaces and the moduli space of point configurations in the projective plane;(v) Lattices in R^n and flows on their moduli space SL_n(R)/\SL_n(Z);(vi) The moduli spaces of iterated polynomials and rational maps in one complex variable; and(vii) The moduli space of vector bundles on a Riemann surface.We focus on problems of rigidity, the statistics and topology of orbits (are all invariant measures algebraic? Is there a spectral gap?), deformation spaces and their compactifications, and ramifications for Diophantine approximation. Can we know the future? Models for planetary motion, evolution of species, climate change and a host of other dynamical systems suggest the answer is yes. But the concerted mathematical study of even the simplest models reveals engines of unpredictability, coexisting with complete knowledge of the underlying laws of change. This project brings analysis, geometry and number theory to bear on the study of mathematical dynamical systems, with the goal of comprehending their core behaviors and what can and cannot be predicted.

该项目涉及动力系统、复分析以及模空间的代数、几何和拓扑之间的相互作用。 本研究的设置包括(i)黎曼曲面M_g的模空间、其复测地线和全纯1-形式丛；(ii)有理多边形中的台球；(iii)实数和复数K3曲面，它们的自同构和由Hodge结构编码的模数；(iv)有理曲面的自同构和 射影平面上点配置的模空间；(v) R^n 中的格子及其模空间 SL_n(R)/\SL_n(Z) 上的流；(vi) 一个复变量中的迭代多项式和有理映射的模空间； (vii) 黎曼面上向量丛的模空间。我们关注刚性问题、轨道的统计和拓扑（所有不变测度都是代数的吗？是否存在谱间隙？）、变形空间及其紧致化以及丢番图近似的分支。我们能知道未来吗？ 行星运动、物种进化、气候变化和许多其他动力系统的模型表明答案是肯定的。 但即使是对最简单模型的协调一致的数学研究也揭示了不可预测性的引擎，与对潜在变化规律的完整了解共存。 该项目将分析、几何和数论应用于数学动力系统的研究，目的是理解其核心行为以及可以预测和不能预测的内容。