Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Realizability and exceptionality of candidate surface branched covers:
  methods and results   Given two closed orientable surfaces, the Hurwitz existence problem asks
whether there exists a branched cover between them having prescribed global
degree and local degrees over the branching points. The Riemann-Hurwitz formula
gives a necessary condition, which was shown to be also sufficient when the
base surface has positive genus. For the sphere one knows that for some data
the cover exists and for some it does not, but the problem is still open in
general. In this paper we will review five different techniques recently
employed to attack it, and we will state the main results they have led to. To
illustrate the techniques we will give five independent proofs of the fact that
there is no branched cover of the sphere over itself with degree 4, three
branching points, and local degrees (2,2), (2,2), and (3,1) over them (despite
the fact that the Riemann-Hurwitz formula is satisfied).

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The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph