Quantum Teichmüller Theory

The classical Teichmüller space of a surface consists of all complex
structures on that surface. The quantum Teichmüller space, as
introduced by Fock, Chekhov and Kashaev, is a certain non- commutative
deformation of the algebra of rational functions on this space. I will
discuss the representation theory of this object, and its connection
to 3-dimensional hyperbolic geometry.