Abstract:

This talk deals with the question on realizability of branch data by branched coverings of the sphere. The Riemann-Hurwitz formula is a necessary condition for realizability but is not sufficient on the sphere. I will introduce this problem and the complete solution provided by Hurwitz in terms of symmetric group theory. I will also discuss about some results obtained by Krzysztof Baranski and Hao Zheng which simplify this algebraic criterion for some classes of branch data. Finally, I will proof that the Riemann-Hurwitz formula is actually a sufficient condition in a very particular case, that shows the existence of a specific class of rational maps.