The Topology and Geometry of Hyperbolic 3-Manifolds

双曲3-流形的拓扑和几何Canary教授将研究标记双曲3-流形同列等价于固定紧化3-流形M的空间AH(M)，由于Ahlfors， Bers, Kra, Marden, Maskit， Sullivan和Thurston的工作，这个空间的内部已经得到了很好的理解。粗略地说，内部的组成部分由拓扑数据枚举，每个组成部分由分析数据参数化。Thurston的端点分层猜想预测整个空间AH(M)中的点可以通过拓扑数据（来自双曲3-流形的标记同胚类型）和几何数据（编码端点的渐近几何）进行分类。Yair Minsky在解决M具有不可压缩边界的Thurston’s Ending Lamination猜想方面取得了重大进展。Canary教授将与Jeff Brock和Yair Minsky合作，完成这一猜想和一些相关猜想的证明。Anderson, Brock, Bromberg, Canary, Holt， McCullough和McMullen最近的工作揭示了关于AH(M)全局拓扑的令人惊讶的新现象。内部的组件可能会碰撞（即有相交的闭包），单个组件可能会碰撞自己。Canary教授建议进一步研究AH(M)的全局拓扑结构。特别地，在攻击结束层积猜想中使用的技术将用于研究结束不变量的连续性。三维流形的研究使用了几何学、拓扑学、动力学和复杂分析的工具。正是这些工具的相互作用赋予了这门学科丰富的风味。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；以熟悉的三维物体（如甜甜圈）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。在19世纪，曲面被分类，并证明所有的曲面都有一个美丽的黎曼度规，它有三种类型：欧几里得、球面或双曲。对固定表面上所有可能的度量的研究始于19世纪，并在拓扑学、复分析、黎曼几何和代数几何中发挥了关键作用。瑟斯顿通过猜想彻底改变了3流形的研究，并在许多情况下证明，每个3流形都可以常规地分解成8种几何类型之一的度量。双曲度量是这类度量中最常见和最不容易理解的一类。Canary教授将研究固定3流形上所有双曲度量的空间。

DMS-0203698Richard CanaryThe topology and geometry of hyperbolic 3-manifoldsProfessor Canary will study the space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M. The interior of this space is well-understood due to work of Ahlfors, Bers, Kra, Marden, Maskit, Sullivan and Thurston. Roughly, the components of the interior are enumerated by topological data and each component is parameterized by analytical data. Thurston's Ending Lamination Conjecture predicts that points in the entire space AH(M) can be classified by topological data (coming from the marked homeomorphism type of the hyperbolic 3-manifold) and geometric data (which encodes the asymptotic geometry of the ends.) Yair Minsky has made substantial progress towards the resolution of Thurston's Ending Lamination Conjecture in the case that M has incompressible boundary. Prof. Canary will collaborate with Jeff Brock and Yair Minsky on a program to complete the proof of this conjecture and a number of related conjectures. Recent work, by Anderson, Brock, Bromberg, Canary, Holt, McCullough and McMullen, has revealed surprising new phenomena concerning the global topology of AH(M). Components of the interior may bump (i.e. have intersecting closures) and individual components may bump themselves. Prof. Canary proposes to further study the global topology of AH(M). In particular, the techniques used in the attack on the Ending Lamination Conjecture will be used to study the continuity of the ending invariants.The study of 3-dimensional manifolds employs tools from geometry, topology, dynamics and complex analysis. It is the interplay of these tools which gives the subject its rich flavor. A 2-dimensionalmanifold or surface is a space which looks locally like 2-dimensionalEuclidean space; examples are given by the surfaces of familiar 3-dimensional objects such as 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. Surfaces were classified in the 19th century, and it was shown that all surfaces admit a beautiful Riemannian metric of one of three types: Euclidean, spherical or hyperbolic. The study of all possible such metrics on a fixed surface was begun in the 19th century and has played a key rolein topology, complex analysis, Riemannian geometry and algebraic geometry. Thurston revolutionized the study of 3-manifolds by conjecturing, and proving in many cases, that every 3-manifoldmay be canonically decomposed into pieces which admit metrics ofone of 8 geometric types. Hyperbolic metrics are the most common and least understood class of such metrics. Prof. Canary will study the space of all hyperbolic metrics on a fixed 3-manifold.