The Teichmüller space of a surface is a fundamental construction in low-dimensional geometry and topology. It is a space parametrizing hyperbolic structures on the surface up to isotopy. The goal of quantum Teichmüller is to construct a quantum deformation of the algebra of functions on the classical Teichmüller space. This course will give an introduction to these ideas and their applications in mathematical physics.
Basic differential geometry and algebraic topology
Topics will be selected from influential papers in quantum Teichmüller theory, possibly including the following:
1. Andersen, J.E. and Kashaev, R. (2014). A TQFT from quantum Teichmüller theory.Communications in Mathematical Physics, 330, 887--934.
2. Bonahon, F. and Wong, H. (2011). Quantum traces for representations of surface groups in SL_2.Geometry & Topology, 15(3), 1569--1615.
3. Fock, V.V. and Chekhov, L.O. (1999). A quantum Teichmüller space.Theoretical and Mathematical Physics, 120(3), 1245--1259.
4. Fock, V.V. and Goncharov, A.B. (2009). The quantum dilogarithm and representations of quantum cluster varieties.Inventiones mathematicae, 175(2), 223--286.
5. Kashaev, R.M. (1998). Quantization of Teichmüller spaces and the quantum dilogarithm.Letters in Mathematical Physics, 43(2), 105--115.
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