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.

从粒子物理学到金融，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表现出这些自然现象的许多特征，例如标志着新状态开始的相变和临界点。这些普遍模式可以通过对所谓的模空间、它们的紧化以及动态不变量的分层的严格研究来揭示。该研究项目吸引了广泛的数学学科，以加深我们对动力系统的理解并提高我们的数学和计算方法。通过缩小搜索范围的理论工具和理想多边形中台球流模拟等实验方法的结合，它的方法已经导致了新的和意想不到的代数体系的发现。格子模空间、黎曼曲面、有理图和其他代数结构表现出丰富的几何形状，通常伴随着刚性和与算术的联系。这些空间还具有动态性质——它们支持自然流动或具有复杂轨道的群体行为，或者对此类行为进行分类。该研究项目从动力学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、完全测地复曲面存在的基础。研究人员还旨在开发 Teichmueller 空间的 L^p 几何，以在 Teichmueller 和 Weil-Petersson 度量之间进行插值并超越 Teichmueller 和 Weil-Petersson 度量（代表 p=1 和 p=2 的情况）。 p=无穷大的情况尤其会导致双曲 3 流形上最尖锐的边界，该边界在圆上。在齐次空间的设置下，研究者计划为合适的开双曲 3 流形建立类 Ratner 刚性定理。此外，在单位圆盘上设置适当的全纯映射时，研究者的目标是开发简单闭合曲线和拉伸映射理论的动力学模拟，增强有理映射和克莱因群之间的字典。