A. Varchenko

activity

19932023

most citedGeometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation

167 citations · 1.9k across the 134 of their papers we have counts for

collaborators

Showing math.AGShow all

49 papers · 1 filter

math.AG2023

Polynomial Solutions Modulo $p^{s}$ of Differential KZ and Dynamical Equations

Pavel Etingof, Alexander Varchenko

We construct polynomial solutions modulo $p^{s}$ of the differential KZ and dynamical equations where $p$ is an odd prime number.

math.AG2021★ 3 cited

Notes on solutions of KZ equations modulo $p^{s}$ and $p$ -adic limit $s \to \infty$

Alexander Varchenko

We consider the KZ equations over $C$ in the case, when the hypergeometric solutions are hyperelliptic integrals of genus $g$ . Then the space of solutions is a $2 g$ -dimensi…

math.AG2020

Derived KZ equations

Vadim Schechtman, Alexander Varchenko

In this paper we strengthen the results of [SV] by presenting their derived version. Namely, we define a "derived Knizhnik - Zamolodchikov connection"\ and identify it with a "deri…

math.AG2020★ 11 cited

The $F_{p}$ -Selberg integral of type $A_{n}$

Richard Rimanyi, Alexander Varchenko

We prove an $F_{p}$ -Selberg integral formula of type $A_{n}$ , in which the $F_{p}$ -Selberg integral is an element of the finite field $F_{p}$ with odd prime number…

math.AG2020

The $F_{p}$ -Selberg Integral

Richard Rimanyi, Alexander Varchenko

We prove an $F_{p}$ -Selberg integral formula, in which the $F_{p}$ -Selberg integral is an element of the finite field $F_{p}$ with odd prime number $p$ of element…

math.AG2020★ 3 cited

Determinant of $F_{p}$ -hypergeometric solutions under ample reduction

Alexander Varchenko

We consider the KZ differential equations over $C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equa…

math.AG2020

The $*$ -Markov equation for Laurent polynomials

Giordano Cotti, Alexander Varchenko

We consider the $*$ -Markov equation for the symmetric Laurent polynomials in three variables with integer coefficients, which is an equivariant analog of the classical Markov equat…

math.AG2020★ 3 cited

An invariant subbundle of the KZ connection mod $p$ and reducibility of $s l^_{2}$ Verma modules mod $p$

Alexander Varchenko

We consider the KZ differential equations over $C$ in the case, when its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same…

math.AG2020

Hypergeometric Integrals Modulo $p$ and Hasse--Witt Matrices

Alexey Slinkin, Alexander Varchenko

We consider the KZ differential equations over $C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equa…

math.AG2019

Positive Populations

Vadim Schechtman, Alexander Varchenko

A positive structure on the varieties of critical points of master functions for KZ equations is introduced. It comes as a combination of the ideas from classical works by G.Luszti…

math.AG2019★ 3 cited

Equivariant quantum differential equation and $q K Z$ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

Giordano Cotti, Alexander Varchenko

In arXiv:1901.02990v1 the equivariant quantum differential equation ( $q D E$ ) for a projective space was considered and a compatible system of difference $q K Z$ equations was introduc…

math.AG2019★ 33 cited

Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

Richárd Rimányi, Andrey Smirnov, Alexander Varchenko +1

Let $X$ be a holomorphic symplectic variety with a torus $T$ action and a finite fixed point set of cardinality $k$ . We assume that elliptic stable envelope exists for $X$ …

math.AG2019★ 30 cited

3d Mirror Symmetry and Elliptic Stable Envelopes

Richárd Rimányi, Andrey Smirnov, Alexander Varchenko +1

We consider a pair of quiver varieties (X;X') related by 3d mirror symmetry, where X =T*Gr(k,n) is the cotangent bundle of the Grassmannian of k-planes of n-dimensional space. We g…

math.AG2019★ 1 cited

Equivariant quantum differential equation, Stokes bases, and K-theory for a projective space

Vitaly Tarasov, Alexander Varchenko

We consider the equivariant quantum differential equation for the projective space $P^{n - 1}$ . We prove an equivariant gamma theorem for $P^{n - 1}$ , which describes the asymptotics o…

math.AG2018

Twisted de Rham Complex on Line and Singular Vectors in $sl_{2}^$ Verma Modules

Alexey Slinkin, Alexander Varchenko

We consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite s…

math.AG2018

Critical points of master functions and mKdV hierarchy of type $C_{n}^{(1)}$

Alexander Varchenko, Tyler Woodruff

We consider the population of critical points, generated from the critical point of the master function with no variables, which is associated with the trivial representation of th…

math.AG2018

Hyperelliptic integrals modulo $p$ and Cartier-Manin matrices

Alexander Varchenko

The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $F_{p}$ with a prime number…

math.AG2017

Potentials of a Frobenius like structure

Claus Hertling, Alexander Varchenko

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame which encompasses families of arrangements. The frame uses the n…

math.AG2017★ 8 cited

Solutions modulo $p$ of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz

Alexander Varchenko

We consider the Gauss-Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallelly to themselves. We…

math.AG2017★ 1 cited

$q$ -Hypergeometric solutions of quantum differential equations, quantum Pieri rules, and Gamma theorem

Vitaly Tarasov, Alexander Varchenko

We describe \, $q$ -hypergeometric solutions of the equivariant quantum differential equations and associated qKZ difference equations for the cotangent bundle $T^{*} F_{λ}$ of a partial…

math.AG2017

Remarks on the Gaudin model modulo $p$

Alexander Varchenko

We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional $s l_{2}$ -modules over the field $F_{p}$ with $p$ elements, where $p$ is a prime number. We d…

math.AG2017

Solutions of KZ differential equations modulo $p$

Vadim Schechtman, Alexander Varchenko

We construct polynomial solutions of the KZ differential equations over a finite field $F_{p}$ as analogs of hypergeometric solutions.

math.AG2017★ 11 cited

Elliptic and K-theoretic stable envelopes and Newton polytopes

R. Rimányi, V. Tarasov, A. Varchenko

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $K$ -theoretic stable envelopes for them and also define a version of the elliptic stable…

math.AG2017

Critical points of master functions and mKdV hierarchy of type $A_{2 n}^{(2)}$

Alexander Varchenko, Tyler Woodruff

We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the…

Polynomial Solutions Modulo ps of Differential KZ and Dynamical Equations

Notes on solutions of KZ equations modulo ps and p-adic limit s→∞

Derived KZ equations

The Fp​-Selberg integral of type An​

The Fp​-Selberg Integral

Determinant of Fp​-hypergeometric solutions under ample reduction

The ∗-Markov equation for Laurent polynomials

An invariant subbundle of the KZ connection mod p and reducibility of sl^2​ Verma modules mod p

Hypergeometric Integrals Modulo p and Hasse--Witt Matrices

Positive Populations

Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

3d Mirror Symmetry and Elliptic Stable Envelopes

Equivariant quantum differential equation, Stokes bases, and K-theory for a projective space

Twisted de Rham Complex on Line and Singular Vectors in sl2​^​ Verma Modules

Critical points of master functions and mKdV hierarchy of type Cn(1)​

Hyperelliptic integrals modulo p and Cartier-Manin matrices

Potentials of a Frobenius like structure

Solutions modulo p of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz

q-Hypergeometric solutions of quantum differential equations, quantum Pieri rules, and Gamma theorem

Remarks on the Gaudin model modulo p

Solutions of KZ differential equations modulo p

Elliptic and K-theoretic stable envelopes and Newton polytopes

Critical points of master functions and mKdV hierarchy of type A2n(2)​

Polynomial Solutions Modulo $p^{s}$ of Differential KZ and Dynamical Equations

Notes on solutions of KZ equations modulo $p^{s}$ and $p$ -adic limit $s \to \infty$

The $F_{p}$ -Selberg integral of type $A_{n}$

The $F_{p}$ -Selberg Integral

Determinant of $F_{p}$ -hypergeometric solutions under ample reduction

The $*$ -Markov equation for Laurent polynomials

An invariant subbundle of the KZ connection mod $p$ and reducibility of $s l^_{2}$ Verma modules mod $p$

Hypergeometric Integrals Modulo $p$ and Hasse--Witt Matrices

Equivariant quantum differential equation and $q K Z$ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

Twisted de Rham Complex on Line and Singular Vectors in $sl_{2}^$ Verma Modules

Critical points of master functions and mKdV hierarchy of type $C_{n}^{(1)}$

Hyperelliptic integrals modulo $p$ and Cartier-Manin matrices

Solutions modulo $p$ of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz

$q$ -Hypergeometric solutions of quantum differential equations, quantum Pieri rules, and Gamma theorem

Remarks on the Gaudin model modulo $p$

Solutions of KZ differential equations modulo $p$

Critical points of master functions and mKdV hierarchy of type $A_{2 n}^{(2)}$