On the Uniformization of Holomorphic Curves

Some directions in the research in Surface Theory ask for a more explicit approach
 to the Uniformization Theorem.  Really explicit examples are rare.
 In this talk I will therefore focus not on the proof by Koebe based on quasi-conformal
 maps, but on Poincaré’s original approach.  The key elements in this approach are
 projective structures, the Schwarzian derivative, and accessory parameters.  At this
 the Schottky uniformization and the work of Zograf and Takhtajan on Conformal Field
 Theory and later work of McMullen relating all this quasi-Fuchsian uniformization come
 into play