Related papers: Origami: real structure, enumeration and quantum m…
A class of quantum analogues of compact symmetric spaces of classical type is introduced by means of constant solutions to the reflection equations. Their zonal spherical functions are discussed in connection with $q$-orthogonal…
Double Hurwitz numbers have at least four equivalent definitions. Most naturally, they count covers of the Riemann sphere by genus g curves with certain specified ramification data. This is classically equivalent to counting certain…
We prove that natural generating functions for enumeration of branched coverings of the pillowcase orbifold are level 2 quasimodular forms. This gives a way to compute the volumes of the strata of the moduli space of quadratic…
We consider composition and division algebras over the real numbers: We note two r\^oles for the group $G_{2}$: as automorphism group of the octonions and as the isotropy group of a generic 3-form in 7 dimensions. We show why they are…
We apply a symbolic approach of the general quadratic decomposition of polynomial sequences - presented in a previous article referenced herein - to polynomial sequences fulfilling specific orthogonal conditions towards two given…
This paper gives one set of axioms for origami constructions, and describes the set of constructible points under these axioms. The determination of the set of cunstructible points for this particular set of axioms is related to Hilbert's…
We propose an explicit and practical algorithm for computing Galois conjugates and irreducible polynomials for special values of modular functions evaluated at CM points associated with imaginary quadratic orders. Our approach builds upon…
The modular form $(azy)_5$ notably appears in one of Igusa's classic structure theorems as a generator of the ring of full modular forms in genus 2, being exhibited by means of a complicated algebraic expression. In this work a different…
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial…
We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula of Goulden, Jackson and Vakil for one part double Hurwitz numbers. Immediate consequences…
In this paper, we collect a number of facts about double Hurwitz numbers, where the simple branch points are replaced by their more general analogues --- completed (r+1)-cycles. In particular, we give a geometric interpretation of these…
Unlike classical modular forms, there is currently no general way to implement the computation of Siegel modular forms of arbitrary weight, level and character, even in degree two. There is however, a way to do it in a unified way. After…
We prove that formal Fourier Jacobi expansions of degree 2 are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension 2, which were defined by Kudla. A second application is…
The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in 7-dimensional projective space. We compute defining polynomials for three versions of this family,…
Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex…
We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group $\mathrm{SU}_{2,2}(\mathcal{O}_K)$ where $K$ is the imaginary-quadratic number field…
We introduce the notion of modular forms, focusing primarily on the group PSL2Z. We further introduce quasi-modular forms, as wel as discuss their relation to physics and their applications in a variety of enumerative problems. These notes…
For any number $m \equiv 0,1 \, (4)$ we correct the generating function of Hurwitz class number sums $\sum_r H(4n - mr^2)$ to a modular form (or quasimodular form if $m$ is a square) of weight two for the Weil representation attached to a…
Suppose q is a complex number of modulus one and different from 1,-1. Let O(R^2_q) be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using operator representations of the *-algebra O(R^2_q) on Hilbert…
We study the homogeneous coordinate rings of real multiplication noncommutative tori as defined by A. Polishchuk. Our aim is to understand how these rings give rise to an arithmetic structure on the noncommutative torus. We start by giving…