Much progress in the study of 3-manifolds has been made by considering the geometric structures they admit. This is nowhere more true than for 3-manifolds which admit a hyperbolic structure. However, in the land of algorithms a more combinatorial approach is necessary, replacing our charts and isometries with finite simplicial complexes that are defined by a finite amount of data. 

In this talk we'll have a look at how in fact one can combine the two approaches, using the geometry of hyperbolic 3-manifolds to assist in this more combinatorial approach. To do so we'll combine tools from Hyperbolic Geometry, Triangulations, and perhaps suprisingly Polynomial Algebra to find explicit bounds on the runtime of an algorithm for comparing Hyperbolic manifolds.

The mapping class group of a surface with boundary acts freely and properly discontinuously on the fatgraph complex, which is a contractible cell complex arising from a cell decomposition of Teichmuller space. We will use this action to get a presentation of the mapping class group in terms of fat graphs, and convert this into one in terms of chord diagrams. This chord slide presentation has potential applications to computing bordered Heegaard Floer invariants for open books with disconnected binding.

Gromov hyperbolicity is a property to metric spaces that generalises the notion of negative curvature for manifolds.
After an introduction about these spaces, we will explain the construction of horocyclic products related to lamplighter groups, Baumslag solitar groups and the Sol geometry.
We will describe the shape of geodesics in them, and present rigidity results on their quasi-isometries due to Farb, Mosher, Eskin, Fisher and Whyte.