We define hypergeometric functions using intersection homology valued in a local system. Topology is emphasized; analysis enters only once, via the Hodge decomposition. By a pull-back procedure we construct special subsets Sπ , derived from Hurwitz spaces, of Deligne-Mostow moduli spaces DM(n, μ). Certain DM(n, μ) are known to be ball quotients, uniformized by hypergeometric functions valued in...

New family of flat potential (Darboux–Egoroff) metrics on the Hurwitz spaces and corresponding Frobenius structures are found. We consider a Hurwitz space as a real manifold. Therefore the number of coordinates is twice as big as the number of coordinates used in the construction of Frobenius structure on Hurwitz spaces found by B. Dubrovin more than 10 years ago. The branch points of a ramifie...

Two integral representations of q-analogues of the Hurwitz zeta function are established. Each integral representation allows us to obtain an analytic continuation including also a full description of poles and special values at non-positive integers of the q-analogue of the Hurwitz zeta function, and to study the classical limit of this qanalogue. All the discussion developed here is entirely ...

Based on an inversion of the Routh table construction, a unimodular characterization of all Hurwitz polynomials is obtained. In parameter space, the Hurwitz polynomials of degree n correspond to the positive 2-tant. The method is then used to construct classes of stable continuous-time delay-difference equations and delay differential equations of neutral type, by a suitable limiting process.

Let Z3 act on C 2 by non-trivial opposite characters. Let X = [C/Z3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials FX and F Y are equal after a change of variables — verifying the Crepant Resolution Conjecture for the pair (X , Y ). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of in...

Let Z3 act on C by non-trivial opposite characters. Let X = [C/Z3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials FX and F are equal after a change of variables — verifying the Crepant Resolution Conjecture for the pair (X , Y ). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of indepe...

In this research-cum-expository article, we aim at presenting a systematic account of some recent developments involving the Riemann Zeta function ζ(s), the Hurwitz (or generalized) Zeta function ζ(s, a), and the Hurwitz-Lerch Zeta function Φ(z, s, a) as well as its various interesting extensions and generalizations. In particular, we begin by looking into the problems associated with the evalu...

Deformations of Dubrovin’s Hurwitz Frobenius manifolds are constructed. The deformations depend on g(g+1)/2 complex parameters where g is the genus of the corresponding Riemann surface. In genus one, the flat metric of the deformed Frobenius manifold coincides with a metric associated with a one-parameter family of solutions to the Painlevé-VI equation with coefficients (1/8,−1/8, 1/8, 3/8) . A...