Yurii Brezhnev (Tomsk State University, Russia)
  Analytic connections on Riemann surfaces.

We give a differentially closed description of the uniformizing representation to the analytical apparatus on Riemann surfaces. Apart from well-known automorphic functions and Abelian differentials it involves construction of the connection objects. We show, in an example of the hyperelliptic curves, how can the connection be explicitly constructed. We study also a relation between classical/traditional `linearly differential' viewpoint (principal Fuchsian equation) and uniformizing representation of the theory.