Related papers: Galois groups of the basic hypergeometric equation…

We study the middle convolution of local systems on the punctured affine line in the setting of singular cohomology and in the setting of \'etale cohomology. We derive a formula to compute the topological monodromy of the middle convolution…

Differential calculus on the quantum quaternionic group GL(1,H$_q$) is introduced.

We evaluate several classes of high weight hypergeometric series via Gamma, polylogarithm and elliptic integrals, mainly through distribution relations.

We study compatible families of four-dimensional Galois representations constructed in the \'{e}tale cohomology of a smooth projective variety. We prove a theorem asserting that the images will be generically large if certain conditions are…

Let $L/K$ be a finite Galois extension whose Galois group $G$ is non-abelian and characteristically simple. Using tools from graph theory, we shall give a closed formula for the number of Hopf-Galois structures on $L/K$ with associated…

Let $C \langle t_1, \dots t_l\rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots ,t_l)$ over an algebraically closed field $C$ of characteristic zero. We develop a lower bound…

All deformations of two dimensional centrally extended Galilei group are classified. The corresponding quantum Lie algebras are found.

These notes are a self-contained introduction to Galois theory, designed for the student who has done a first course in abstract algebra.

We prove that any semi-simple representation of the Galois group of a number field coming from geometry appears as a subquotient of the ring of regular functions on the pro-algebraic completion of the fundamental group of the projective…

We formulate and prove a weighted version of Zariski's hyperplane section theorem on the topological fundamental groups of the complements of hypersurfaces in a projective space. As an application, we calculate fundamental groups of the…

Let $H$ be a skew field of finite dimension over its center $k$. We solve the Inverse Galois Problem over the field of fractions $H(X)$ of the ring of polynomial functions over $H$ in the variable $X$, if $k$ contains an ample field.

We show that a linear algebraic group is the Galois group of a parameterized Picard-Vessiot extension of k(x), x' = 1, for certain differential fields k, if and only if its identity component has no one dimensional quotient as a linear…

We introduce new hypergeometric series expansions of the solutions to the general Heun equation. The form of the Gauss hypergeometric functions used as expansion function differs from that used before. We derive three such expansions and…

The survey contains a brief description of the ideas, constructions, results, and prospects of the theory of hypergroups and generalized translation operators. Representations of hypergroups are considered, being treated as continuous…

The $p$-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group $G = \text{Gal} (k^{nr,2}/k)$ for $k = \mathbb{Q}(\sqrt{d})$ with $d = -445, -1015, -1595, -2379$. In…

Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.

In this paper, we study some extended hypergeometric functions from matrix point of view. We have given the integral representations of these matrix functions. Finally, we obtain some generating function relations using fractional…

We study the structure of abelian subgroups of Galois groups of function fields of surfaces.

We classify isoparametric hypersurfaces in complex hyperbolic spaces.

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…