We present a review of the spin Hurwitz numbers, which count ramified coverings with structures. It is known that they are related to characters Sergeev group and Q Schur functions. This allows one put whole story into context matrix models integrable hierarchies. The generating functions numbers $$\tau ^\pm $$ hypergeometric -functions BKP hierarchy; we their fermionic realization. cut-and-joi...

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M , the Hurwitz constant of M . It measures how well all g...

Hurwitz spaces are spaces of pairs S, f where S is a Riemann surface and f : S → ̂ C a meromorphic function. In this work, we study 1-dimensional Hurwitz spaces Hp of meromorphic p-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of p − 1 /2 transpositions and the monodromy group is the dihedral group ...

This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group Γ = 〈 x, y | x = y = 1 〉 of size at least 6. This characterization may be viewed as a discrete analogue of the description of Hurwitz groups as finite quotients of the (2, 3, 7)-triangle group in the context...

In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri–Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivial exponential taufunction. In fact, in the double scaling Kashaev limit (the ...

Hurwitz transformations are defined as specific automorphisms of a CayleyDickson algebra. These transformations generate quadratic and nonquadratic forms. We investigate here the Hurwitz transformations corresponding to Cayley-Dickson algebras of dimensions 2m = 2, 4 and 8. The Hurwitz transformations which lead to quadratic forms are discussed from geometrical and Lie-algebraic points of view....

Fixing an arbitrary point p ∈ CP and a triple (g, d, `) of nonnegative integers satisfying the inequality g ≤ (d+l−1 2 ) − (l 2 ) , we associate a natural Hurwitz number to the (open) Severi-type varietyWg,d,` consisting of all reduced irreducibke plane curves of degree d + l with genus g and having an ordinary singularity of order l at p (the remaining singular points of such curves being usua...

We present the first polynomial uniform random sampling algorithm for simple branched coverings of the sphere by itself of degree n. More precisely, our algorithm generates in linear time increasing quadrangulations, which are equivalent combinatorial structures. Our result is based on the identification of some canonical labelled spanning trees, and yields a constructive proof of a celebrated ...

Authors, define and establish a new subclass of harmonic regular schlicht functions (HSF) in the open unit disc through the use of the extended generalized Noor-type integral operator associated with the ξ-generalized Hurwitz-Lerch Zeta function (GHLZF). Furthermore, some geometric properties of this subclass are also studied.