Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
Alex Eskin,Andrei Okounkov +1 more
TLDR
In this paper, the authors 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.Abstract:
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.read more
Citations
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Gromov-Witten theory, Hurwitz theory, and completed cycles
TL;DR: In this article, an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles was established.
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Billiards and Teichmüller curves on Hilbert modular surfaces
TL;DR: In this paper, an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two, which lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication.
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Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants
TL;DR: In this paper, it was shown that having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one.
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Dynamics of SL2(ℝ) Over Moduli Space in Genus Two
TL;DR: In this paper, the authors classify orbit closures and invariant measures for the natural-action of SL2(R) on UM2, the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two.
References
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Symmetric functions and Hall polynomials
TL;DR: In this paper, the characters of GLn over a finite field and the Hecke ring of GLs over finite fields have been investigated and shown to be symmetric functions with two parameters.
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On quantum gauge theories in two dimensions
TL;DR: In this article, the authors studied two-dimensional quantum Yang-Mills theory from three points of view: (i) by standard physical methods; (ii) by relating it to the large-k limit of three dimensional Chern-Simons theory and two dimensional conformal field theory; and (iii) by connecting its weak coupling limit to the theory of Reidemeister-Ray-Singer torsion.
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Connected components of the moduli spaces of Abelian differentials with prescribed singularities
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