Description
Abstract: The focus of this talk le is the study of moduli spaces of representations of fundamental groups of surfaces S with boundaries with values in GL(,n C). In absence of marked points on the boundary, this moduli space is realized in many equivalent ways: as the moduli space of linear local systems on S, as the moduli space of representations of the fundamental groupoid, as the moduli space of monodromy data and as character variety. By adding marked points to the boundary of S in order to capture irregular singularities, the Betti moduli space has been generalized in several ways by many authors. Although it is clear that these different approaches describe essentially the same spaces of mathematical objects, exactly how they fit together has not yet been established. In this talk, I will develop a categorical framework that allows for a clear and systematic definition of the decorated Betti moduli space designed to specialize to the different points of view encountered in the literature.