Abstract

In this talk I'll describe a Lie groupoid based approach to the study of flat connections with logarithmic singularities on a hypersurface. Flat connections on the affine line with logarithmic singularity at the origin are equivalent to representations of a groupoid associated to the exponentiated action of C. I'll describe a canonical Jordan-Chevalley decomposition for these representations, and show how this leads to a functorial classification. Flat connections on a general complex manifold with logarithmic singularities along a hypersurface are equivalent to representations of a twisted fundamental groupoid. By using a Morita equivalence, the category of logarithmic flat connections can be localized to the normal bundle of the hypersurface. I'll explain how this can be used to prove a functorial Riemann-Hilbert correspondence for logarithmic flat connections.