Complex Dynamics and Moduli Spaces

复杂动力学和模空间

基本信息

批准号：
1903764
负责人：
Curtis McMullen
金额：
$ 100万
依托单位：
Harvard University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1903764&HistoricalAwards=false
关键词：
Complex Dynamics Moduli Spaces

项目摘要

From particle physics to finance, from evolution to climate change, the world is full of dynamical systems. Simple algebraic transformations already exhibit many of the features of these natural phenomena, such as phase transitions and tipping points that signal the onset of new regimes. These universal patterns may be revealed through the rigorous study of moduli spaces, their compactifications and their stratifications by dynamical invariants. This project appeals to a broad range of mathematical disciplines to both deepen our understanding of dynamical systems and to sharpen our mathematical and computational methods. Its methods have already led to the discover of new and unexpected algebraic regimes, through a combination of theoretical tools that narrow the domain of search, and experimental methods such as the simulation of bouncing molecules in idealized chambers.A central issue in science, from biology to mathematics, is the study and classification of the wide variations that can take place in a single recognized species. The moduli spaces constructed by Ahlfors, Bers and Mumford in the 1960s, give this kind of classification for compact Riemann surfaces with a fixed genus g and number of marked points n. Today the study of moduli spaces is the meeting ground for disciplines ranging from arithmetic geometry to string theory. In dimension three, Riemann surfaces are replaced by hyperbolic 3-manifolds. This project aims to expand the frontiers of our understanding of both moduli spaces and hyerbolic 3-manifolds from the perspective of dynamical systems, unified by the action of SL2(R) in both regimes. It draws on methods ranging from complex analysis to low--dimensional topology and renormalization. The concerted study of specific examples, assisted by computer visualization and algebraic manipulation, also plays a central role of in this research.The PI and his coworkers have recently discovered new totally geodesic curves and surfaces in the moduli spaces for (g,n) = (4,0), (1,3), (1,4) and (2,1). Their methods also reproduce most previously known examples of these rare and beautiful objects. A central goal of this project is to put forth a unified construction of all known examples, and investigate the prospects for obtaining a complete classification. The project will also address problems concerning totally geodesic planes in open hyperbolic 3-manifolds, the complexity of closed loops on surfaces, the arithmetic underlying billiards in polygons, moduli spaces of rational maps, and connection between twisted 1-forms and 3-manifolds that fiber over the circle. The training of graduate students to become active and independent researchers forms a central part of this project. The field of complex dynamics and moduli spaces is one where concrete problems and examples abound, and yet some of the deepest insights and methods of modern mathematics can be brought to bear. It invites the participation of experts from a broad range of fields. This project will continue to foster the growth of an open, diverse and active research network.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

从粒子物理学到金融，从进化到气候变化，世界都充满了动态系统。简单的代数转换已经表现出这些自然现象的许多特征，例如信号是新制度开始的相变和倾斜点。这些通用模式可以通过对模量空间的严格研究，它们的压缩及其分层通过动态不变性来揭示。该项目吸引了广泛的数学学科，以加深我们对动态系统的理解，并提高我们的数学和计算方法。它的方法已经导致发现了新的和意外的代数制度，这是通过缩小搜索领域的理论工具的结合以及实验方法，例如理想化的钱伯斯中弹跳分子的模拟。科学中的一个中心问题，从生物学到数学的核心问题，是对广泛的知识变化的研究和分类，可以实现单一认可物种的广泛变异。 1960年代由Ahlfors，Bers和Mumford构建的模量空间，为紧凑型Riemann表面提供了这种分类，具有固定的属G和标记点n的数量。如今，对模量空间的研究是学科从算术几何到弦理论等学科的见面。在第三维中，Riemann表面被双曲线3个manifolds取代。该项目旨在从动态系统的角度来扩大我们对模量空间和卫生3个manifolds的理解的前沿，这是由SL2（r）在这两个制度中的作用统一的。它利用从复杂分析到低维拓扑结构和重新归一化的方法。在计算机可视化和代数操纵的协助下，对特定示例进行的协同研究在这项研究中也起着核心作用。PI和他的同事最近在（G，n）=（4,0），（1,3），（1,4）和（1,4）和（2,1）中发现了（g，n）=（g，n）=（g，n）=（g，n）=（g，n）=（g，n）=（g，n）=（g，n）=（g，n）=（g，n）的表面。他们的方法还复制了这些稀有和美丽物体的最先前已知的例子。该项目的一个核心目标是提出所有已知示例的统一结构，并调查获得完整分类的前景。该项目还将解决有关在开放双曲线3个manifolds中完全大地测量平面的问题，封闭环的复杂性，多边形中的算术基础台球，理性图的模量空间，以及扭曲的1型1forms和3个纤维之间的连接。对研究生的培训成为活跃和独立的研究人员，构成了该项目的核心部分。复杂的动力学和模量空间的领域是具体问题和例子比比皆是的，但是一些现代数学的最深刻的见解和方法可以带来。它邀请了来自广泛领域的专家的参与。该项目将继续促进开放，多样化和活跃的研究网络的增长。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。