(Higher) Teichmüller spaces and beyond

The Teichmüller space of a surface parametrizes (marked) conformal structures. It covers the moduli space of Riemann surfaces and carries many interesting structures in its own right. The uniformization theorem allows to identify the Teichmüller space with the space of hyperbolic structures, and to embed into the space of representations of the fundamental group of the surface into SL(2,R).

Higher Teichmüller spaces are generalizations of Teichmüller space in the context of Lie groups of higher rank such as SL(n,R). They are related to Higgs bundles, bounded cohomology, dynamics, as well as cluster algebras or total positivity. The talk will provide an introduction to higher Teichmüller spaces and showcase some of the connections.