Complex Dynamics and Moduli Spaces

复杂动力学和模空间

• 批准号: 1608432
• 负责人: Curtis McMullen
• 金额: $ 51.67万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1608432&HistoricalAwards=false
• 关键词: Complex; Dynamics; Moduli; Spaces

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 so-called moduli spaces, their compactifications, and their stratifications by dynamical invariants. This research 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 discovery 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 billiard flows in idealized polygons. Moduli spaces of lattices, Riemann surfaces, rational maps and other algebraic structures exhibit rich geometry, often accompanied by rigidity and a connection with arithmetic. These spaces also have a dynamical nature -- they support natural flows or group actions with complicated orbits, or they classify such actions. This research project investigates moduli spaces from a dynamical and geometric perspective. In the setting of Riemann surfaces, the project aims to reveal the mechanisms within dynamics, algebraic geometry, and number theory that underlie the existence of unexpected, recently-discovered primitive, totally geodesic complex surfaces in moduli space. The investigator also aims to develop the L^p geometry of Teichmueller space, to interpolate between and go beyond the Teichmueller and Weil-Petersson metrics (which represent the cases p=1 and p=2). The case p=infinity in particular should lead to the sharpest bounds on the hyperbolic 3-manifold that fiber over the circle. In the setting of homogeneous spaces, the investigator plans to establish Ratner-like rigidity theorems for suitable open hyperbolic 3-manifolds. Additionally, in the setting of proper holomorphic maps on the unit disk, the investigator aims to develop a dynamical analogue of the theory of simple closed curves and stretch maps, enhancing the dictionary between rational maps and Kleinian groups.

从粒子物理到金融，从进化到气候变化，这个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，例如标志着新政权开始的相变和临界点。这些普遍模式可以通过严格研究所谓的模空间，它们的紧化和它们的动态不变量分层来揭示。这个研究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，又提高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，通过结合理论工具，缩小搜索领域，和实验方法，如模拟台球流在理想的多边形。格、黎曼曲面、有理映射和其他代数结构的模空间表现出丰富的几何形状，通常伴随着刚性和与算术的联系。这些空间也具有动态性质——它们支持具有复杂轨道的自然流动或群体行动，或者它们对这些行动进行分类。本研究项目从动态和几何角度研究模空间。在黎曼曲面的背景下，该项目旨在揭示动力学、代数几何和数论中的机制，这些机制是模空间中意想不到的、最近发现的原始的、完全测地的复杂曲面存在的基础。研究者还致力于发展Teichmueller空间的L^p几何，在Teichmueller和Weil-Petersson度量（代表p=1和p=2的情况）之间进行插值并超越它们。特别是p=∞的情况下，应该会导致纤维在圆上的双曲3流形上的最尖锐的界限。在齐次空间的背景下，研究者计划建立合适的开放双曲3流形的类拉特纳刚性定理。此外，在单位盘上适当全纯映射的背景下，研究者旨在建立简单闭曲线理论和伸缩映射理论的动态模拟，增强有理映射和Kleinian群之间的字典。