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

Canary教授的研究关注的是 双曲三维流形的几何和拓扑。 在 过去，他已经探索了知道一个 双曲3-流形在拓扑上是驯服的，即同胚于 紧致三维流形的内部 他特别研究了 谱几何和拓扑驯服的动力学 双曲三维流形 一份申请证明了阿尔福斯 拓扑驯服双曲三维流形测度猜想 在这个项目中，他打算研究的问题是， 双曲三维流形是拓扑驯服的。 这个猜想是 每一个双曲三维流形， 基本群是拓扑驯服的。 他还打算 研究在给定条件下所有双曲结构的空间 三歧管 一个n流形是这样一个对象， 是一个在n维空间中看起来像球的邻域。 例如，单位球面（在三维空间中）是 2-流形，因为关于每个点都有一个小邻域 看起来像是平面上的圆盘 一种自然的方式， 流形的出现是作为几个方程的解集 变量 例如，单位球面是 特别简单的方程（x^2 + y^2 + z^2 = 1）。 上的度量 流形M是测量M上点之间距离的一种方法。 流形可以允许许多不同的度量。 比如说 单位球和苹果的表面是拓扑的 相同的流形，但它们具有不同的度量。 研究 流形上的可能度量类型在以下方面有许多应用： 数学和物理。 研究者将研究以下指标： 3-流形，使得关于任何点， 可以被识别的邻域（通过 保持距离）与双曲三维空间中的球。 这些 度量称为双曲度量。 双曲度量 2-流形最近引起了物理学家的极大兴趣， 尤其是弦理论家。

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.