Related papers: On double Hurwitz numbers in genus 0
This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and…
Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and…
We study the structures of ordinary simple Hurwitz numbers and monotone Hurwitz numbers with varying genus. More precisely, we prove that when the ramification type is fixed and the genus is treated as a variable, the connected monotone…
Hurwitz numbers with completed cycles are standard Hurwitz numbers with simple branch points replaced by completed cycles. In fact, simple branch points correspond to completed $2$-cycles. Okounkov and Pandharipande have established the…
We obtain bivariate asymptotics for one part monotone Hurwitz numbers in high genus (i.e. as both the size and the genus go to infinity). To do so, we start with a linear recurrence for these numbers obtained by Do and Chaudhuri. Then, we…
As the real counterpart of double Hurwitz number, the real double Hurwitz number depends on the distribution of real branch points. We consider the problem of asymptotic growth of real and complex double Hurwitz numbers. We provide a lower…
We compute the number of (weak) equivalence classes of branched covers from a surface of genus g to the sphere, with 3 branching points, degree 2k, and local degrees over the branching points of the form (2,...,2), (2h+1,1,2,...,2),…
We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of…
We compute the first explicit polynomials with Galois groups $G=P\Gamma L_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb{Q}(t)$. Furthermore we compute the first examples of totally real polynomials with Galois groups…
The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distribution of a quantum Bose gas with vanishing fugacity. We compute the leading term of the partition function and the quantum…
This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov-Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov-Witten theory, with a particular…
In recent years, monotone double Hurwitz numbers were introduced as a naturally combinatorial modification of double Hurwitz numbers. Monotone double Hurwitz numbers share many structural properties with their classical counterparts, such…
The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to…
We give uniform formulas for the number of full reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus-0 Hurwitz numbers. This paper is the…
We provide two examples of complex homogeneous quadratic polynomials P on Banach spaces of the form l_1(I). The first polynomial P has both separable and nonseparable maximal zero subspaces. The second polynomial P has the property that…
Let L be the zero set of a nonconstant monic polynomial with complex coefficients. In the context of constructive mathematics without countable choice, it may not be possible to construct an element of L. In this paper we introduce a notion…
This paper aims at extending the criterion that the quasi-stability of a polynomial is equivalent to the total nonnegativity of its Hurwitz matrix. We give a complete description of functions generating doubly infinite series with totally…
Hurwitz spaces which parametrize branched covers of the line play a prominent role in inverse Galois theory. This paper surveys fifty years of works in this direction with emphasis on recent advances. Based on the Riemann-Hurwitz theory of…
In this paper, we give results that partially prove a conjecture which was discussed in our previous work (arXiv:1307.4991). More precisely, we prove that as $n\to \infty,$ the zeros of the polynomial$${}_{2}\text{F}_{1}\left[…
Going beyond the studies of single and double Hurwitz numbers, we report some progress towards studying Hurwitz numbers which correspond to ramified coverings of the Riemann sphere involving three nonsimple branch points. We first prove a…