Extension des théories de Teichmüller-Thurston – ETTT

The study of moduli spaces of geometric structures on surfaces is an important theme in the development of mathematics over the last decades (Teichmüller theory, etc). Some important and exciting developments have happened more recently (including in France). The deal with new approaches of some already widely studied questions, but also with wide generalizations of classical mathematical objects which exhibit remarkable mathematical properties. The project is composed of a number of inter-related taks which are all different ways of generalizing or strenghtening Teichmüller-Thurston theories. The goals are to better understand: - representations of surface groups in SL(n,R) or in more general Lie groups, including the geometry and the quantization of their moduli spaces, - the "higher quasifuchsian representations" (representations of surfaces groups in SL(n,C), - branched projective structures on surfaces and their relations with the Riemann-Hilbert problem and with complex curves in the quotients of PSL(2,C), - the quantizations of moduli spaces of representations of surface groups and their relations to invariants of 3-manifolds, - the relations between representations of surface groups and integrable systems, - the renormalized volume of hyperbolic 3-manifolds and its applications to Teichmüller theory, - AdS geometry and its application to the dynamics of earthquakes, - the geometry of quasifuchsian or convex co-compact hyperbolic 3-manifolds, including those with "particles" (cone singularities along infinite lines), - combinatorial models and the topology at the boundary of representation spaces.