Combinatorics, Models, and Bounds in Hyperbolic Geometry

曲面上曲线的复合体中的“有界组合数学”概念控制着各种环境中的几何图形。理解和界定曲面上的简单闭曲线如何投影到子曲面上，为构造双曲三维流形的模型、分类Weil-Petersson测地线和阐明变形空间边界上的精细拓扑结构提供了工具。事实上，三维流形上双曲结构的存在和唯一性很少提供关于它们的几何特征以及它们与流形的拓扑性质的联系的信息。通过他对双Lipschitz模型定理(与Canary和Minsky的PI)的使用的扩展，PI将探索任意闭双曲3-流形的几何模型，并将其结构与流形的组合特征联系起来。这个正在进行的项目与合作伙伴Minsky，Namazi和Souto将为Heegaard分裂产生显式模型，并为由于无限粘合有界组合数学而产生的3-流形产生一个新的弱几何化。此外，PI将应用粗略的方法通过曲线复形及其相关的“层次路径”来研究映射类群，以了解Teichmueller空间上的Weil-Petersson度量的大尺度结构，尽管进行了许多研究，但其大尺度几何仍然是一个神秘的基本对象。最后，PI将展示双曲三维流形的变形空间的进一步特征，推广了我们与Bromberg，Canary和Minsky对变形空间的局部拓扑的研究，并在曲线复形的背景下对变形空间的中心紧性定理进行了推广研究。在数学上，动力系统的研究试图用简单的术语描述混沌现象。有时动力系统会表现出一种刚性，在这种情况下，微小的调整或扰动不会影响系统的长期行为。在理解我们自己的三维宇宙以及三维空间可以具有什么样的结构的背景下，结构产生了这些刚性动力系统。格里沙·佩雷尔曼最近解决著名的庞加莱猜想的工作确保了这项研究几乎适用于所有三维空间。当一个空间是刚性的时，一个人可以通过“粗略”的信息，通过所谓的“模型”来完全理解它。在与R.Canary和Y.Minsky的PI最近的一个结果中，这样的模型被用来分类所有在一定意义上是驯服的无限体积的、连续的负曲线型或“双曲型”三维空间。分类结果解决了威廉·瑟斯顿长期以来的一个猜想，并为开发以前被认为已被理解的流形上更详细和完整的几何图景打开了大门。为基本研究所有三维空间的代数和拓扑性质以及这些性质之间的相互关系奠定了基础。

The notion of "bounded combinatorics" in the complex of curves on a surface controls geometry in a variety of settings. Understanding and bounding how simple closed curves on a surface can project to subsurfaces provides tools to construct models for hyperbolic 3-manifolds, classify Weil-Petersson geodesics, and elucidate the fine topological structure in boundaries of deformation spaces. Indeed, the existence and uniqueness of hyperbolic structures on 3-manifolds gives little information about their geometric features and their connection to topological properties of the manifold. Through his expansion of the use of the bi-Lipschitz Model Theorem (of the PI with Canary and Minsky), the PI will explore geometric models for arbitrary closed hyperbolic 3-manifolds and connect their structure to combinatorial features of the manifold. This ongoing project with collaborators Minsky, Namazi and Souto will produce explicit models for Heegaard splittings, and a new weak geometrization for 3-manifolds arising as infinite gluings with bounded combinatorics. Further, the PI will apply coarse methods in the study of the mapping class group via the curve complex and its associated "hierarchy paths" to understand the large scale structure of the Weil-Petersson metric on Teichmueller space, a fundamental object whose large scale geometry remains mysterious despite many investigations. Finally, the PI will exhibit further features of the deformation space of a hyperbolic 3-manifold, generalizing our study of the local topology of deformation spaces with Bromberg, Canary, and Minsky, and undertaking generalized studies of central compactness theorems for deformation spaces in the context of the curve complex.In mathematics, the study of dynamical systems seeks to describe chaotic phenomena in simple terms. Sometimes dynamical systems can exhibit a kind of rigidity, where a small tweak or perturbation does not affect the long term behavior of the system. In the context of understanding our own three-dimesnsional universe and what kind of structures three-dimensional spaces can have, structures give rise to these rigid dynamical systems. The recent work of Grisha Perelman solving the famous Poincare Conjecture has ensured this study applies to virtually all three-dimensional spaces. When a space is rigid, one can understand it completely via "coarse" information, via so-called "models." In a recent result of the PI with R. Canary and Y. Minsky, such models were used to classify all constantly negatively curved, or "hyperbolic" three-dimensional spaces of infinite volume that are tame in a certain sense. The classification result solved a long-standing conjecture of William Thurston, and opened the door to developing a more detailed and complete picture of geometries on manifolds previously considered understood. The groundwork is in place for a fundamental investigation of algebraic and topological properties of all spaces of 3-dimensions and how these properties interrelate.