Quantum Teichmuller and Thurston theories
We use the graph description of Teichmuller spaces of Riemann surfaces with holes due to Penner and Fock to attain an explicit description of coordinates on the Teichmuller and moduli spaces. We use the quantum version of this construction to describe the geodesic algebra and extend it to the algebra of laminations. We then obtain the variant of quantum laminations (Chekhov, Penner, 2003). I will hopefully discuss possible relations to 2+1 gravity and to matrix models.