The Topology and Geometry of Hyperbolic 3-Manifolds

在过去的三年中，双曲3-流形的几何和变形理论这一主题发生了相当大的变化和进步。马登的驯服猜想、瑟斯顿的终层猜想、伯斯-沙利文-瑟斯顿的密度猜想和阿尔福斯的测度猜想都已成立。此外，双曲型3流形的变形理论也被揭示出比以前预想的更为复杂。强大的新技术已经被引入到该领域，包括模型流形、曲线复合体的几何形状、钻井定理和末端缩减。Canary教授建议利用这些新技术来提高我们对双曲3-流形几何的理解，进一步阐明仍然神秘的双曲3-流形变形理论，并为该领域的研究探索新的方向。Canary打算继续他对三维双曲流形的拓扑和几何的研究。二维流形或曲面是一个局部看起来像二维欧几里得空间的空间；我们以熟悉的三维物体（如足球或甜甜圈）的表面为例。类似地，三维流形是一个局部看起来像三维欧几里得空间的空间。黎曼度规是一种测量流形中距离和角度的方法。例如，我们生活的宇宙是一个带有黎曼度规的3流形。因此，除其他原因外，研究三维流形及其几何形状是自然而重要的。我们推测三维流形可以被常规地分解成8种几何类型之一的碎片。双曲流形是这类几何流形中最常见也是最不容易理解的一类。近年来，在理解这种流形方面取得了很大进展，Canary教授建议在此基础上进一步深化这一理解。

The subject of geometry and deformation theory of hyperbolic 3-manifolds has seen considerable change and advancement over the last three years. Marden's Tameness Conjecture, Thurston's Ending Lamination Conjecture, the Bers-Sullivan-Thurston Density Conjecture and Ahlfors' Measure Conjecture have all been established. Moreover, the deformation theory of hyperbolic 3-manifolds has been revealed to be more complicated than previously expected. Powerful new techniques have been introduced to the field, including model manifolds, the geometry of the curve complex, drilling theorems, and end reductions. Prof. Canary proposes to use the new techniques to improve our understanding of the geometry of hyperbolic 3-manifolds, to further illuminate the still mysterious deformation theory of hyperbolic 3-manifolds and to explore new directions for research in the field.Prof. Canary proposes to continue his study of the topology and geometry of 3-dimensional hyperbolic manifolds. A 2-dimensional manifold or surface is a space which looks locally like 2-dimensional Euclidean space; examples are given by the surfaces of familiar 3-dimensional objects such as footballs or doughnuts. Similarly, a 3-manifold is a space that looks locally like 3-dimensional Euclidean space. A Riemannian metric is a way of measuring distances and angles in a manifold. For example, the universe we live in is a 3-manifold with a Riemannian metric. For this reason, among others, the study of 3-dimensional manifolds and their geometries is natural and important. It is conjectured that 3-dimensional manifolds can be canonically decomposed into pieces of one of 8 geometric types. The hyperbolic manifolds are the most common and least understood class of such geometric manifolds. In recent years, much progress has been made in understanding such manifolds and Prof. Canary proposes to build on and deepen that understanding.