I'm studying the relations between hyperbolic geometry in dimension 2 and anti-de Sitter (AdS) geometry in dimension 3. For example, there exists an equivalence between minimal Lagrangian diffeomorphisms between hyperbolic surfaces and maximal surfaces in Globally Hyperbolic AdS space-times. In my thesis, I extended these relations to the case of hyperbolic surfaces with cone singularities: it that case, minimal Lagrangian maps are related to maximal surfaces in space-time with particles. More generally, I'm interested in geometric structures associated to Anosov representations of surface groups into semi-simple Lie groups. I also study relations between quantum Teichmüller theory and Topological Quantum Field Theory.
