In this talk I will discuss the question of what is the maximum volume of a hyperbolic polyhedron with a given combinatorics. This question naturally leads, in order to obtain a satisfying answer, to consider polyhedra with vertices « beyond infinity ». I will further explain how this question arises naturally in the study of quantum invariants of 3-manifolds (and especially the volume conjecture).

https://indico.math.cnrs.fr/event/8656/