Glimpses of Mathematics, Now and Then: A Celebration of Karen Uhlenbeck's 80th birthday

Abstract: The Teichm\"uller space of a compact surface S is realized as a connected component of the space of conjugacy classes of representations of the fundamental group into the Lie group PSL(2,R). This component is distinguished by the fact that it consists entirely of representations which are injective with discrete image. A higher Teichm\"uller space is by definition a component of the space of conjugacy classes of representations of the fundamental group of S into a semisimple Lie group G whose rank is at least 2 which consists entirely of injective representations with discrete image. In this talk I will describe how a Higgs bundle version of a Slodowy slice can be used to parameterize all known (and conjecturally all) higher Teichm\"uller spaces.