Some tame fundamental groups and their representation varieties

In this talk, based on joint work with St. Wiedmann (Göttingen), certain representation varieties of tame fundamental groups are studied. The two cases under consideration are the tame fundamental groups of projective algebraic curves over finite fields, resp. Riemann surfaces. The techniques used are appropriate versions of the Langlands correspondence, resp. theorems of Hitchin and Atiyah–Bott on moduli spaces of Higgs and ordinary vector bundles.