Related papers: Towards the topological recursion for double Hurwi…
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 define a new class of enumerative invariants called $k$-leaky double Hurwitz descendants, generalizing both descendant integrals of double ramification cycles and the $k$-leaky double Hurwitz numbers introduced in previous work of…
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…
We interpret the degrees which arise in Tevelev's study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for…
We give an alternative proof of the Hurwitz existence problem for branched covers of $\mathbb{P}^1$ in the case where the number of ramification points equals the number of branch points, that is, where all the ramification profiles are of…
Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz…
We provide a direct correspondence between the $b$-Hurwitz numbers with $b=1$ from \cite{ChapuyDolega}, and twisted Hurwtiz numbers from \cite{TwistedHurwitz}. This provides a description of real coverings of the sphere with ramification on…
We study double Hurwitz numbers in genus zero counting the number of covers $\CP^1\to\CP^1$ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise…
The construction of hypergeometric 2D Toda $\tau$-functions as generating functions for quantum Hurwitz numbers is extended here to multispecies families. Both the enumerative geometrical significance of these multispecies quantum Hurwitz…
We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H.…
The second author and Norbury initiated the enumeration of lattice points in the Deligne-Mumford compactifications of moduli spaces of curves. They showed that the enumeration may be expressed in terms of polynomials, whose top and bottom…
We consider multi-matrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed…
In this paper, we aim to provide an accessible survey to various formulae for calculating single Hurwitz numbers. Single Hurwitz numbers count certain classes of meromorphic functions on complex algebraic curves and have a rich geometric…
In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety. We are going to use their…
We introduce and study a semigroup structure on the set of irreducible components of the Hurwitz space of marked coverings of a complex projective curve with given Galois group of the coverings and fixed ramification type. As application,…
Hurwitz numbers count genus g, degree d covers of the projective line with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain…
A geometric quantization using the topological recursion is established for the compactified cotangent bundle of a smooth projective curve of an arbitrary genus. In this quantization, the Hitchin spectral curve of a rank $2$ meromorphic…
We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of…
An elliptic orbifold is the quotient of an elliptic curve by a finite group. In 2001, Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular…
We study a generalization of the Harish-Chandra - Itzykson - Zuber integral to tensors and its expansion over trace-invariants of the two external tensors. This gives rise to natural generalizations of monotone double Hurwitz numbers, which…