|Date||Dec 15, 2015|
I will first explain the usual Teichmuller space of a Riemann surface in terms of the representation variety of the surface for PSL(2,R). Then I will replace PSL(2,R) with other Lie groups to define higher Teichmuller spaces. For groups PGL(n) and SL(n), I will briefly explain the coordinate system of Fock-Goncharov, which has the structure of the cluster algebra. If time allows, I will mention the Poisson structure and a quantization of this coordinate system.