Mathematical Sciences: The Topology and Geometry of Hyperbolic 3-Manifolds

9626578金丝雀教授将探索有关双曲3-流形和克莱因群的形变理论的各种猜想。其中几个猜想是由Marden的温和性猜想引起的，该猜想预言每个具有有限生成的基本群的双曲3-流形都是拓扑驯服的，即同胚到紧致的3-流形的内部。金丝雀教授之前已经证实，拓扑驯服对双曲三维流形的几何有很强的影响。他还将研究克莱因群序列的代数极限和几何极限之间的关系。这项研究的灵感来自于瑟斯顿的结束分层猜想，该猜想给出了所有双曲3-流形的猜想分类。3-流形是一个数学空间，使得关于任何一点都有一个邻域，它可以用三维空间中的球来标识。人们可以想象通过将3维块粘合在一起来构建3维流形。当然，这种粘合通常必须是一种抽象的粘合，而不是一种可以在三维空间中完成的粘合。例如，考虑通过将单位立方体的顶部粘贴到底部、将正面粘贴到背面、将左侧粘贴到右侧而获得的3-流形。黎曼度量是测量三维流形中距离和角度的一种方法。例如，我们生活的世界是一个具有黎曼度规的三维流形。在20世纪70年代的S中，维利亚姆·瑟斯顿猜想，每个3-流形都可以以规范的方式被分割，使得每个片断都有8种几何类型之一的黎曼度量。他证明了他对大类三维流形的猜想。包含黎曼度量的八种几何类型中的七种的3-流形是完全分类的，并且被很好地理解。包含第八类度量的3-流形称为双曲流形，目前在几何和拓扑学领域是一个非常感兴趣和活跃的课题。金丝雀教授正在研究双曲三维流形的拓扑和几何之间的关系。***

9626578 Canary Professor Canary will explore a variety of conjectures concerning hyperbolic 3-manifolds and the deformation theory of Kleinian groups. Several of these conjectures are motivated by Marden's tameness conjecture, which predicts that every hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, i.e., homeomorphic to the interior of a compact 3-manifold. Professor Canary has previously established that topological tameness has strong consequences for the geometry of a hyperbolic 3-manifold. He will also study the relationship between the algebraic limit and the geometric limit of a sequence of Kleinian groups. This study is inspired by Thurston's ending lamination conjecture, which provides a conjectural classification of all hyperbolic 3-manifolds. A 3-manifold is a mathematical space such that about any point there is a neighborhood which can be identified with a ball in 3-dimensional space. One can imagine building 3-dimensional manifolds by gluing together 3-dimensional blocks. Of course, this gluing would have to be an abstract gluing in general, not one which could be done in 3-dimensional space. For example, consider the 3-manifold obtained by taking the unit cube and gluing the top to the bottom, the front to the back, and the left side to the right side. A Riemannian metric is a way of measuring distances and angles in a 3-manifold. For instance, the world we live in is a 3-manifold with a Riemannian metric. In the 1970's, Wiliam Thurston conjectured that every 3-manifold can be cut up, in a canonical way, so that each piece has a Riemannian metric of one of 8 geometric types. He proved his conjecture for large classes of three-manifolds. The 3-manifolds which admit seven of the eight geometric types of Riemannian metrics are completely classified and well-understood. 3-manifolds that admit the eighth type of metric are called hyperbolic and are currently a subject of intense interest and act ivity in the fields of geometry and topology. Professor Canary is investigating the relationship between the topology and the geometry of hyperbolic 3-manifolds. ***