Abstract:
In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points.  The Weyl problem, in the hyperbolic setting,  asks for isometric embedding of surfaces of curvature at least -1 in to the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture,  the Weyl problem and the work of Alexandrov and Thurston on convex surfaces, and discrete uniformization of polyhedral surfaces.  This is a joint work with Tianqi Wu.