Abstract

Higher Rank Teichmüller theory studies connected components of character varieties of fundamental groups of topological surfaces in semisimple Lie groups that only consist of injective representations with discrete image. This very active field is undergoing rapid and exciting development that I will motivate and put in perspective. I will then focus on joint work with Beyrer, Guichard, Labourie and Wienhard in which we prove a longstanding conjecture that such components arise in conjunction with the Lie algebraic notion of Theta-positivity, and discover some geometric features of these representations bearing strong analogies with holonomies of hyperbolic structures on surfaces. Time permitting I will also discuss applications of real algebraic geometry in theirs study, that I developed with Burger, Iozzi and Parreau.