Spaces of Hyperbolic 3-Manifolds

拟共形形变空间的拓扑、射影结构的分类、无限生成Kleinian群以及几何有限Kleinian群的密度猜想的新方法等。例如，我们生活的空间是一个三维流形。大多数三维流形是双曲的。也就是说它们有一个常曲率度量等于-1。在这个项目中，主要研究者将研究携带整个双曲度量族的3-流形。这些变形空间具有非常复杂的结构，类似于更著名的Mandelbrot集。首席研究员还将研究表面上的复杂投影结构。射影结构与双曲三维流形密切相关，关于一个对象的结果往往会导致关于另一个对象的结果。

Some of the topics that will be studied include: the topology of quasiconformal deformation spaces, classification of projective structures, infinitely generated Kleinian groups and new approaches to the density conjecture for geometrically finite Kleinian groups.A 3-manifold is a mathematical object of fundamental interest. For example the space we live in is a 3-manifold. Most 3-manifolds are hyperbolic. That is they have a metric of constant curvature equal to -1. In this project the principal investigator will study 3-manifolds that carry whole families of hyperbolic metrics. These deformation spaces have an extremely complicated structure that resembles the more well known Mandelbrot set. The principal investigator will also study complex projective structures on surfaces. Projective structures are intimately related to hyperbolic 3-manifolds and results about one object often lead to results about the other.