The Topology and Geometry of Hyperbolic 3-Manifolds

建议：DMS-9971554首席研究员：Richard D.Canary摘要：Canary教授建议研究双曲三维流形的形变空间。在他的第一个项目中，他将研究等价于固定紧致三维流形M的双曲三维流形同伦空间AH(M)的全局拓扑。它的内部分支由拓扑数据枚举，每个分支由解析数据参数化。在M有不可压缩边界的情况下，Anderson，Canary和McCullough给出了内部的两个分支何时有相交闭包的完整刻画。金丝雀建议进一步研究相交轨迹的拓扑结构，从而更好地理解空间。通过Sullivan词典，对AH(M)的研究类似于复杂动力学中的Mandelbrot集的研究。Canary还建议研究几何有限的双曲三维流形的拟共形变形空间的边界，以及曲面群的任意几何极限的结构。金丝雀研究3-流形的拓扑和几何之间的关系。3-流形是一个数学空间，使得在任何一点上都有一个看起来像三维空间中的区域的区域。黎曼度量是测量三维流形中距离和角度的一种方法。一旦有了流形上的黎曼度量，就可以谈论它的几何了。例如，我们生活的世界是一个具有黎曼度规的三维流形。双曲度量是一种特别好的黎曼度量，自19世纪首次发现以来，拓扑学家和地质学家对它进行了广泛的研究。最近，物理学家们一直在讨论我们宇宙的几何模型，这些模型具有双曲度规。为自己的宇宙模型选择度量值，关系到我们的宇宙是在膨胀还是在收缩的问题。金丝雀的大部分研究涉及同一拓扑流形上不同双曲度量空间的研究。

Proposal: DMS-9971554Principal Investigator: Richard D. Canary Abstract: Professor Canary proposes to study deformation spaces of hyperbolic 3-manifolds. In his first project, he will will study the global topology of the space AH(M) of hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M. The components of its interior are enumerated by topological data and each component is parameterized by analytic data. In the case that M has incompressible boundary, Anderson, Canary and McCullough have given a complete characterization of when two components of the interior have intersecting closures. Canary proposes to further investigate the topology of the intersection locus and thus develop a better understanding of the space. This study of AH(M) is analogous, via the Sullivan dictionary, to the study of the Mandelbrot set in complex dynamics. Canary also proposes to study boundaries of quasiconformal deformation spaces of geometrically finite hyperbolic 3-manifolds, as well as the structure of arbitrary geometric limits of surface groups. Canary studies the relationship between the topology and the geometry of a 3-manifold. A 3-manifold is a mathematical space such that about any point there is a region which looks like a region in 3-dimensional space. A Riemannian metric is a way of measuring distances and angles in a 3-manifold. Once one has a Riemannian metric on a manifold one can talk about its geometry. For example, the world we live in is a 3-manifold with a Riemannian metric. A hyperbolic metric is an especially nice type of Riemannian metric, which has been extensively studied by topologists and geometers since it was first discovered in the nineteenth century. Recently, physicists have been discussing models for the geometry of our universe which have hyperbolic metrics. The choice of metric for one's model of the universe is related to the issue of whether our universe is expanding or contracting. Much of Canary's research involves the study of spaces of different hyperbolic metrics on the same topological manifold.