Event № 139
Classical Teichmüller theory is concerned with the study of (marked) Riemann surfaces. Due to the uniformization theorem, the Teichmüller space of a surface can be also described as the space of (marked) hyperbolic structures on a given topological surface S. A hyperbolic structure on S is governed by a discrete embedding of the fundamental group of S into the Lie group PSL(2,R). Higher Teichmüller theory concerns the study of more complicated geometric structures on surfaces, which are governed by discrete embeddings of the fundamental group of S into more general Lie groups, such as PSL(n,R) or Sp(2n,R).
I will give an introduction to higher Teichmüller theory, review some of the recent results and discuss current and future challenges.