Spaces of Hyperbolic 3-Manifolds

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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.