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

加那利教授将探讨关于双曲3-流形和Kleinian群的变形理论的各种猜想。这些猜想中有几个是由Marden的驯服猜想引起的，该猜想预测了每一个具有有限生成基本群的双曲3-流形在拓扑上是驯服的，即与紧化3-流形的内部同纯。Canary教授之前已经证实，拓扑驯服性对双曲3流形的几何有很强的影响。他还将研究Kleinian群序列的代数极限和几何极限之间的关系。本研究的灵感来自瑟斯顿的结束层压猜想，该猜想提供了所有双曲3-流形的推测分类。三维流形是一个数学空间，使得在任何一点上都有一个可以与三维空间中的球识别的邻域。人们可以想象通过将三维块粘合在一起来构建三维流形。当然，这种粘合必须是一种抽象的粘合，而不是在三维空间中进行的粘合。例如，考虑通过取单位立方体并将顶部粘合到底部，前部粘合到后部，左侧粘合到右侧获得的3-歧管。黎曼度规是一种在三维流形中测量距离和角度的方法。例如，我们生活的世界是一个带有黎曼度规的3流形。在20世纪70年代，威廉·瑟斯顿（william Thurston）推测，每个3流形都可以被切割，以一种规范的方式，这样每个部分都有一个8种几何类型之一的黎曼度规。他在三流形的大类别中证明了他的猜想。承认黎曼度量的8种几何类型中的7种的3流形是完全分类和很好理解的。承认第八种度量的3流形被称为双曲流形，目前是几何和拓扑学领域中非常感兴趣和活跃的主题。Canary教授正在研究双曲型3流形的拓扑结构和几何结构之间的关系。＊＊＊

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. ***