Given a homeomorphism f between Riemann surfaces, Teichmueller's theorem states that there exists a unique quasiconformal map h, homotopic to f, which minimizes the quasiconformal dilatation. Moreover, there exists polygon representing the two Riemann surfaces such that h is an affine map which stretches the polygon in one direction and contracts it in the other. In this talk, we will ask a similar question, given a topological branched cover f between Riemann surfaces, does there exist a unique quasiregular map h homotopic to f which minimizes the quasiregular dilatation? To answer that question, we will rephrase the question in terms of Teichmueller geometry and study some particularly interesting subspaces of Teichmueller space. Speaker(s): Maxime Scott (Indiana University)