Complex Dynamics and Moduli Spaces

从粒子物理到金融，从进化到气候变化，世界充满了动力系统。简单的代数变换已经表现出这些自然现象的许多特征，例如相变和预示新制度开始的临界点。这些普适模式可以通过严格研究模空间及其紧致化和动力学不变量的层次化来揭示。这个项目吸引了广泛的数学学科，既加深了我们对动力系统的理解，也加强了我们的数学和计算方法。它的方法已经导致了新的和意想不到的代数制度的发现，通过缩小搜索范围的理论工具和实验方法的组合，例如模拟理想化房间中的弹跳分子。从生物学到数学，科学的一个中心问题是研究和分类在单个已知物种中可能发生的广泛变化。由Ahlfors，Bers和Mumford在20世纪60年代构造的模空间给出了具有固定亏格g和标记点数n的紧致Riemann曲面的这种分类。今天，模空间的研究是从算术几何到弦论的多个学科的交汇点。在三维空间中，黎曼曲面被双曲三维流形所代替。这个项目旨在从动力系统的角度扩展我们对模空间和双曲3-流形的理解的前沿，通过SL2(R)在这两个区域中的作用来统一。它借鉴了从复杂分析到低维拓扑和重整化的各种方法。在计算机可视化和代数处理的辅助下，对具体实例的协同研究也是这项研究的核心。最近，Pi和他的同事们在模空间中发现了新的全测地曲线和曲面，其中(g，n)=(4，0)，(1，3)，(1，4)和(2，1)。他们的方法还复制了这些稀有而美丽的物体的大多数先前已知的例子。这个项目的一个中心目标是对所有已知的例子提出一个统一的结构，并调查获得一个完整分类的前景。该项目还将解决关于开双曲3-流形中的全测地平面、曲面上闭环的复杂性、多边形中的台球的算术、有理映射的模空间以及环上扭曲的1-形式与3-流形之间的联系等问题。培养研究生成为积极和独立的研究人员是这一项目的核心部分。在复杂动力学和模空间领域，具体的问题和例子比比皆是，但现代数学的一些最深刻的见解和方法可以被运用。它邀请来自广泛领域的专家参加。该项目将继续促进开放、多样化和积极的研究网络的发展。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.