The Teichmüller space of a surface that admits a hyperbolic structure can be equipped with the Weil-Petersson metric. The Weil-Persson Ending Lamination is a candidate …
Broadly, topology is the study of shapes. In this thesis we specifically study surfaces through graphs which are defined by characteristics of the surface. The …
This thesis is devoted to understanding incompressible surfaces properly embedded in a hyperbolic four-punctured sphere bundle. The ideas are drawn from previous works of Floyd, …
The focus of this dissertation is on the invertibility of certain topological summary statistics for metric objects. The first set of results concern persistence diagrams …