Related papers: Combinatorics of pruned Hurwitz numbers
Various generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the Eynard-Orantin topological recursion, an infinite-order differential equation called a…
We introduce stable tropical curves and use these to count covers of the $p$-adic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the…
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…
We consider $d$-fold branched coverings of the projective plane $\mathbb{RP}^2$ and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz…
The main goal of this paper is to introduce the notion of a primitive form for a generic family of Hurwitz covers of $\mathbb{P}^1$ with a fixed ramification profile over infinity. We prove that primitive forms are in one-to-one…
We use tropical and nonarchimedean geometry to study the moduli space of genus $0$ stable maps to $\mathbb{P}^1$ relative to two points. This space is exhibited as a tropical compactification in a toric variety. Moreover, the fan of this…
We present the geometry lying behind counting twin prime polynomials in $\mathbb{F}_q[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties…
We consider the bit complexity of computing Chow forms and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational…
In this note we provide a new partial solution to the Hurwitz existence problem for surface branched covers. Namely, we consider candidate branch data with base surface the sphere and one partition of the degree having length two, and we…
We show that the combinatorial Lefschetz number is a topological invariant. This is an important result in itself; in order to point it out, we will also work here several relevant consequences in different directions. The first of them is…
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…
Hurwitz (1887) defined a continued fraction algorithm for complex numbers which is better behaved in many respects than a more "natural" extension of the classical continued fraction algorithm to the complex plane would be. Although the…
In contrast to all other known Ramanujan-type congruences, we discover that Ramanujan-type congruences for Hurwitz class numbers can be supported on non-holomorphic generating series. We establish a divisibility result for such…
We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types.…
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…
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…
Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann…
In [14] we found the large genus asymptotics of Hurwitz numbers for the Riemann sphere with a fixed number of general profiles and some (2,1^{d-2}) profiles. In this paper, motivated from [3], we generalize these results to Hurwitz numbers…
We reduce the calculation of the simplest Hodge integrals to some sums over decorated trees. Since Hodge integrals are already calculated, this gives a proof of a rather interesting combinatorial theorem and a new representation of…
The theory of quasirandomness has greatly expanded from its inaugural graph theoretical setting to several different combinatorial objects such as hypergraphs, tournaments, permutations, etc. However, these quasirandomness variants have…