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

Canary教授的工作涉及双曲3-流形的几何和拓扑之间的关系。在过去，他探索了知道双曲3-流形在拓扑上是驯服的，即同胚于紧致3-流形内部的后果。特别是，他研究了拓扑驯服的双曲3-流形的谱几何和动力学。其中一个应用是对拓扑温和的双曲3-流形的Ahlfors测度猜想的证明。在这个项目中，他打算研究哪些双曲3-流形在拓扑上是温和的。猜想是每一个具有有限生成基本群的双曲3-流形都是拓扑驯服的。他还打算研究给定的3-流形上所有双曲结构的空间。N维流形是这样一种对象，即在n维空间中，每个点附近都有一个看起来像球的邻域。例如，单位球(在三维空间中)是2-流形，因为每个点周围都有一个看起来像平面中的圆盘的小邻域。流形产生的一种自然方式是作为多变量方程的解集。例如，单位球是一个特别简单的方程(x^2+y^2+z^2=1)的解集。流形M上的度量是度量M上的点之间距离的一种方法。流形可能包含许多不同的度量。例如，单位球面和苹果表面在拓扑上是相同的流形，但它们具有不同的度量。研究流形上可能度量的类型在数学和物理中有许多应用。研究人员将研究3-流形上的度量，使得关于任何一点都有一个小邻域，它可以(通过保持距离的标识)与双曲3-空间中的球识别。这些指标被称为双曲线指标。最近，2-流形上的双曲度量引起了物理学家，特别是弦理论家的极大兴趣。

Professor Canary's work concerns the relationship between the geometry and the topology of hyperbolic 3-manifolds. In the past he has explored the consequences of knowing that a hyperbolic 3-manifold is topologically tame, i.e. homeomorphic to the interior of a compact 3-manifold. In particular, he studied the spectral geometry and the dynamics of topologically tame hyperbolic 3-manifolds. One application was a proof of Ahlfors' measure conjecture for topologically tame hyperbolic 3-manifolds. In this project he intends to study the question of which hyperbolic 3-manifolds are topologically tame. The conjecture is that every hyperbolic 3-manifold with finitely generated fundamental group is topologically tame. He also intends to study the space of all hyperbolic structures on a given 3-manifold. An n-manifold is an object such that about every point there is a neighborhood which looks like a ball in n-dimensional space. For example, the unit sphere (in 3-dimensional space) is a 2-manifold, since about each point there is a small neighborhood which looks like a disk in the plane. One natural way in which manifolds arise is as solution sets of equations in several variables. For example, the unit sphere is the solution set of a particularly simple equation (x^2 + y^2 + z^2 = 1). A metric on a manifold M is a way of measuring distances between points on M. A manifold may admit many different metrics. For example, the unit sphere and the surface of an apple are topologically the same manifold, but they have different metrics. Studying the types of possible metrics on a manifold has many applications in mathematics and physics. The investigator will study metrics on 3-manifolds such that about any point there is a small neighborhood which may be identified (by an identification which preserves distances) with a ball in hyperbolic 3-space. These metrics are called hyperbolic metrics. Hyperbolic metrics on 2-manifolds have recently been of great interest to physicists, in particular, to string theorists.