Quantum hyperbolic geometry

I will discuss emerging connections between hyperbolic
geometry and topological quantum field theory. More precisely, the
combinatorics of a manifold of dimension 2 and 3 gives rise to
interesting algebraic objects, which are non-commutative deformations
of the algebra of functions on the space of hyperbolic structures on
the manifold. Combining geometric and algebraic data provides new
invariants for these manifolds.