Thurston's Skinning Map and Curves on Surfaces

The ‘deformation space' of a given geometric structure on a fixed smooth manifold is a major theme in low-dimensional geometry. In this thesis we present work on two problems that fit into such a framework for 2 and 3-dimensional hyperbolic manifolds. The first concerns the behavior of the ‘skinning map’ associated to the family of infinite-volume hyperbolic structures on the interior of a 3-manifold with boundary, and the second concerns counting ‘maximal complete 1-systems’ on a closed surface.