Related papers: Generic Solutions of Equations Involving the Modul…

The purpose of this paper is to provide answers to some questions raised in a paper by Kaneko and Koike about the modularity of the solutions of a differential equations of hypergeometric type. In particular, we provide a number-theoretic…

We first propose a generalization of the image conjecture [Z3] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent…

We give a method to find quartic Heisenberg invariant equations for Kummer varieties and we give some explicit examples. From these equations for g-dimensional Kummer varieties one obtains equations for the moduli space of g+1-dimensional…

We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we…

In this paper we study generalized J-inner matrix valued functions which appear as resolvent matrices in various indefinite interpolation problems. Reproducing kernel indefinite inner product spaces associated with a generalized J-inner…

The Andr\'e-Pink-Zannier conjecture concerns the intersection of subvarieties and the generalized Hecke orbit of a given point in mixed Shimura varieties. It is part of the Zilber-Pink conjecture. In this paper we focus on the universal…

Deligne proved that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g>1 must be integral or half integral. We give a different proof for this. It uses Mennicke's result that subgroups of…

We consider a matrix nonlinear partial differential equation that generalizes Heisenberg ferromagnet equation. This generalized Heisenberg ferromagnet equation is completely integrable with a linear bundle Lax pair related to the…

In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their arguments. More precisely, when $J_{K,0}$…

We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with point masses…

The Deligne-Simpson problem is formulated like this: give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset SL(n,{\bf C})$ or $c_j\subset sl(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples of…

We establish new operational formulae of Burchnall type for the complex disk polynomials (generalized Zernike polynomials). We then use them to derive some interesting identities involving these polynomials. In particular, we establish…

We give a new numerical criterion in the spirit of GIT for existence of solutions to inverse Hessian equations, including in particular the J-equation. Our criterion is formulated in terms of stability of pairs in the sense of Paul. To that…

Self-adjoint Dirac systems and subclasses of canonical systems, which generalize Dirac systems are studied. Explicit and global solutions of direct and inverse problems are obtained. A local Borg-Marchenko-type theorem, integral…

In this article we give a general approach to the following analogue of Shafarevich's conjecture for some polarized algebraic varieties; suppose that we fix a type of an algebraic variety and look at families of such type of varieties over…

We prove the meromorphy of solutions for a wide class of ordinary differential equations. These equations are given by invariant manifolds of non-linear partial differential equations integrable by the inverse scattering method. Some higher…

Given a simplicial complex whose vertices are labeled with positive integers, one can associate a vector configuration whose corresponding toric variety is the Zariski closure of a hierarchical model. We classify all the vertex-weighted…

We formulate a generalization of Vojta's conjecture in terms of log pairs and variants of multiplier ideals. In this generalization, a variety is allowed to have singularities. It turns out that the generalized conjecture for a log pair is…

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for…

Modular categories are important algebraic structures in a variety of subjects in mathematics and physics. We provide an explicit, motivated and elementary definition of a modular category over a field of characteristic 0 as an equivalence…