Complex Dynamics and Moduli Spaces

从粒子物理到金融，从进化论到气候变化，世界充满了动力系统。 简单的代数变换已经展示了这些自然现象的许多特征，例如相变和临界点，它们标志着新制度的开始。这些普遍的模式可以通过模空间的严格研究，它们的紧化和动力学不变量的分层来揭示。 这个项目呼吁广泛的数学学科，既加深我们对动力系统的理解，又提高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政权的发现，通过理论工具的结合，缩小了搜索的领域，和实验方法，如模拟在理想化的房间里弹跳的分子。科学的一个中心问题，从生物学到数学，是研究和分类的广泛变化，可以发生在一个单一的公认的物种。 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等学科的交汇点。在三维中，黎曼曲面被双曲三维流形所取代。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，并通过SL 2（R）在这两个领域中的作用来统一。它借鉴了从复分析到低维拓扑和重整化的方法。PI和他的同事最近在模空间（g，n）=（4，0），（1，3），（1，4）和（2，1）中发现了新的全测地曲线和曲面。他们的方法还重现了这些罕见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提出一个统一的建设所有已知的例子，并调查获得一个完整的分类的前景。该项目还将解决有关开放双曲3流形中的全测地平面，表面上闭合回路的复杂性，多边形中台球的算术基础，有理映射的模空间以及扭曲1形式和3流形之间的连接的问题。培养研究生成为积极和独立的研究人员是该项目的核心部分。复杂动力学和模空间领域是一个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承担。它邀请来自广泛领域的专家参加。该项目将继续促进一个开放、多样化和活跃的研究网络的发展。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.