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严格研究所谓的模空间，它们的紧化，以及它们的动力学不变量的分层来揭示。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政权的发现，通过理论工具的结合，缩小了搜索的域，和实验方法，如模拟台球流在理想化的多边形。格的模空间、黎曼曲面、有理映射和其他代数结构表现出丰富的几何学，通常伴随着刚性和与算术的联系。这些空间也有一个动态的性质-它们支持自然流或具有复杂轨道的群体行为，或者它们对这些行为进行分类。本研究计画从动力学与几何学的观点探讨模空间。在黎曼曲面的设置中，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存在意想不到的，最近发现的原始，完全测地线复曲面的基础。研究者还旨在发展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）之间插值并超越它们。特别是p=无穷大的情况，应该会导致在圆上纤维化的双曲三维流形上的最尖锐的边界。在齐次空间的背景下，研究者计划建立合适的开双曲三维流形的Ratner-like刚性定理。此外，在单位圆盘上的适当全纯映射的设置中，研究者的目标是开发简单闭曲线和拉伸映射理论的动力学模拟，增强有理映射和Kleinian群之间的字典。

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.