Related papers: Schinzel's Problem: Imprimitive covers and the mon…

We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.

In an earlier paper we propose an approach to the unitarizability problem in the case of classical groups over a p-adic field of characteristic zero based on cuspidal reducibility points. We have reduced earlier the unitarizability for…

We show that braided Cherednik algebras introduced by the first two authors are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups $G(m,p,n)$, when $m$ is even. This gives a new construction of mystic…

For every nonconstant monic polynomial $g \in \mathbb{Z}[X]$, let $\mathfrak{M}(g)$ be the set of positive integers $m$ for which there exist an integer linear recurrence $(s_n)_{n \geq 0}$ having characteristic polynomial $g$ and a…

This is the first of a series of papers to construct the deformation theory of the form Schr\"odinger equation, which is related to a section-bundle system $(M,g,f)$, where $(M,g)$ is a noncompact complete K\"ahler manifold with bounded…

This paper is a subsequent paper of math.RT/0607673. Here we consider the irreducible components of Springer fibres (or orbital varieties) for two-column case in GL}_n. We describe the intersection of two irreducible components, and…

Let ${\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, smooth curves. For a family $C\to S$ of such curves over a connected base and a geometric point $\xi$ on $S$, the associated monodromy representation is…

Chiral conformal blocks in a rational conformal field theory are a far going extension of Gauss hypergeometric functions. The associated monodromy representations of Artin's braid group capture the essence of the modern view on the subject,…

Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of order $q$. When $q$ is small, we make further assumptions on $G$. Then we determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal representations…

Given a finite nonempty set of primes S, we build a graph $\mathcal{G}$ with vertex set $\mathbb{Q}$ by connecting x and y if the prime divisors of both the numerator and denominator of x-y are from S. In this paper we resolve two…

We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve $\cal Y$ with bi-degree (2,2) in a product of projective lines ${\Bbb P}^{1} \times {\Bbb P}^{1}$. We calculate two…

For a subvariety of a smooth projective variety, consider the family of smooth hypersurfaces of sufficiently large degree containing it, and take the quotient of the middle cohomology of the hypersurfaces by the cohomology of the ambient…

We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive…

Let $G$ be a finite non-regular primitive permutation group on a set $\Omega$ with point stabiliser $G_{\alpha}$. Then $G$ is said to be extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus…

Primitive equivalence of graphs and matrices was used by Enomoto, Fujii and Watatani to classify Cuntz-Krieger algebras of 3x3 irreducible matrices. In this paper it is shown that the definition of primitive equivalence can be simplified…

Motivated by our results in "Polish Models and Sofic Entropy," we define modified version of the independence tuples for sofic entropy developed by Kerr and Li. These modified version essentially require that the independence sequences give…

Formulating a Schubert problem as the solutions to a system of equations in either Pl\"ucker space or in the local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation…

We consider the sums of the form $$ S=\sum_{x=1}^{N} \exp\big((ax+b_1g_1^x+... +b_rg_r^x)/p \big) $$, where $p$ is prime and $g_1,..., g_r$ are primitive roots $\pmod p$. An almost forty years old problem of L. J. Mordell asks to find a…

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite…

Our main goal is to study the freeness of Sch\"utzenberger groups defined by primitive substitutions. Our findings include a simple freeness test for these groups, which is applied to exhibit a primitive invertible substitution with…