Hyperbolic manifolds from a contact geometry perspective

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After Thurston's work we know that the world of 3-manifolds comes in different flavors, depending on the geometry of the manifolds we are looking at. From a different perspective, we know that every 3-manifold admits a contact structure. There is a classic dichotomy in the realm of contact structures: they can be tight or overtwisted. The former ones are the most interesting from a mathematical perspective. The classification of tight contact structures on 3-manifolds has been at the center of the mathematical research for the last 20 years. The first well understood examples were the 3-sphere, the solid torus and lens spaces. There are now well established techniques to deal with Seifert manifolds, but the world of tight contact structure on hyperbolic manifolds is still largely unexplored. In this PhD project the student will start tackling the problem of classification of tight contact structures on hyperbolic manifolds.