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In recent years there has been a vivid interaction between mathematics and physics involving representations of surface groups, in particular the Hitchin component and the space of positive representations introduced by Fock and Goncharov. These spaces are examples of higher Teichmüller spaces. In this talk I will discuss a unifying framework that includes all family of higher Teichmüller spaces. A particular example will be the space of maximal representations into the symplectic group, which can also be viewed as a non-commutative Teichmüller space.

Lunch follows the talk.