• branched covering
• Hurwitz
• short partition
• surface

The Hurwitz existence problem asks what branching data can actually be realized by a branched covering between surfaces. The two main approaches we discuss in this thesis involve, respectively, monodromy and dessins d'enfant. The monodromy approach consists in finding appropriate representations of the fundamental group of a surface in the symmetric group, and it can be successfully employed to fully solve the existence problem in non-positive Euler characteristic. Dessins d'enfant, on the other hand, provide a topological and combinatorial approach for showing realizability or exceptionality of branching data on the sphere, especially in the case of n=3 branching points.
In this thesis we introduce these two methods in detail, and we apply them to a few well-known instances of the Hurwitz existence problem. Later, we focus on branching data with a "short" (i.e. of length 2) partition: we develop some genus-reducing "combinatorial moves" which operate on dessins d'enfant, and we systematically apply them to provide the first full solution to the existence problem for data with a short partition.
Finally, we briefly touch on a computational approach developed by Zheng, and we extend his results by computing the complete list of exceptional data of degree d<30. We find that every branching datum with n=23 and n=29 is realizable, thereby providing further evidence of the well-known prime-degree conjecture.