Asymptotics of numbers of branched coverings of a torus and volumes of   moduli spaces of holomorphic differentials
  We compute the asymptotics of the number of connected branched coverings of a torus as their degree goes to infinity and the ramification type stays fixed. These numbers are equal to the volumes of the moduli spaces of pairs (curve, holomorphic differential) with fixed multiplicities of zeros of the differential and have several applications in ergodic theory. 
Alex Eskin and Andrei Okounkov