Victor Matveevich Buchstaber
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View article: Hyperelliptic sigma functions and the Kadomtsev–Petviashvili equation
Hyperelliptic sigma functions and the Kadomtsev–Petviashvili equation Open
View article: Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action
Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action Open
Plücker coordinates define the $T^n$-equivariant embedding $p : G_{n,2}\to \C P^{N}$ of a complex Grassmann manifold $G_{n,2}$ into the complex projective space $\C P^{N}$, $N=\binom{n}{2}-1$ for the canonical $T^n$-action on $G_{n,2}$ and…
View article: $n$-Valued Groups, Kronecker Sums, and Wendt's Matrices
$n$-Valued Groups, Kronecker Sums, and Wendt's Matrices Open
The article presents results on the well-known problem concerning the structure of integer polynomials $p_n(z; x, y)$, which define multiplication laws in $n$-valued groups $\mathbb{G}_n$ over the field of complex numbers $\mathbb{C}$. We …
View article: Some Open Mathematical Problems on Fullerenes
Some Open Mathematical Problems on Fullerenes Open
Fullerenes are hollow carbon molecules where each atom is connected to exactly three other atoms, arranged in pentagonal and hexagonal rings. Mathematically, they can be combinatorially modeled as planar, 3-regular graphs with facets compo…
View article: Random eigenvalues of nanotubes
Random eigenvalues of nanotubes Open
The hexagonal lattice and its dual, the triangular lattice, serve as powerful models for comprehending the atomic and ring connectivity, respectively, in graphene and carbon (p,q)-nanotubes . The chemical and physical attributes of these t…
View article: Kolmogorov widths, Grassmann manifolds and unfoldings of time series
Kolmogorov widths, Grassmann manifolds and unfoldings of time series Open
Problems in Kolmogorov's theory of widths and the theory of unfoldings of time series are considered. These theories are related by means of the theory of extremal problems on the Grassmann manifolds $G(n,q)$ of $q$-dimensional linear subs…
View article: Random eigenvalues of graphenes and the triangulation of plane
Random eigenvalues of graphenes and the triangulation of plane Open
We analyze the numbers of closed paths of length on two important regular lattices: the hexagonal lattice (also called graphene in chemistry) and its dual triangular lattice. These numbers form a moment sequence of specific random v…
View article: Algebraic $2-$valued group structures on $\mathbb P^1$, Kontsevich-type polynomials, and multiplication formulas, I
Algebraic $2-$valued group structures on $\mathbb P^1$, Kontsevich-type polynomials, and multiplication formulas, I Open
The theory of a two-valued algebraic group structure on a complex plane and complex projective line is developed. In this theory, depending on the choice of the neutral element, the local multiplication law is given by the Buchstaber polyn…
View article: Some open mathematical problems on fullerenes
Some open mathematical problems on fullerenes Open
Fullerenes are hollow carbon molecules where each atom is connected to exactly three other atoms, arranged in pentagonal and hexagonal rings. Mathematically, they can be combinatorially modeled as planar, 3-regular graphs with facets compo…
View article: Moduli space of weighted pointed stable curves and toric topology of Grassmann manifolds
Moduli space of weighted pointed stable curves and toric topology of Grassmann manifolds Open
We relate the theory of moduli spaces $\overline{\mathcal{M}}_{0,\mathcal{A}}$ of stable weighted curves of genus $0$ to the equivariant topology of complex Grassmann manifolds $G_{n,2}$, with the canonical action of the compact torus $T^n…
View article: Polynomial dynamical systems associated with the KdV hierarchy
Polynomial dynamical systems associated with the KdV hierarchy Open
In 1974, S.P. Novikov introduced the stationary n-equations of the Korteweg–de Vries hierarchy, namely the n-Novikov equations. These are associated with integrable polynomial dynamical systems, with polynomial 2n integrals, in ℂ3n. In thi…
View article: Random eigenvalues of nanotubes
Random eigenvalues of nanotubes Open
The hexagonal lattice and its dual, the triangular lattice, serve as powerful models for comprehending the atomic and ring connectivity, respectively, in \textit{graphene} and \textit{carbon $(p,q)$--nanotubes}. The chemical and physical a…
View article: Sultan Nazhmudinovich Askhabov (on his 70-th birthday)
Sultan Nazhmudinovich Askhabov (on his 70-th birthday) Open
The article is devoted to the 70-th anniversary of Sultan Nazhmudinovich Askhabov, Doctor of Physical and Mathematical Sciences, Professor of the Kadyrov Chechen State University, a well-known specialist in the field of nonlinear integral,…
View article: Todd Polynomials and Hirzebruch Numbers
Todd Polynomials and Hirzebruch Numbers Open
In 1956 Hirzebruch found an explicit formula for the denominators of the Todd polynomials, which was proved later in his joint work with Atiyah. We present a new formula for the Todd polynomials in terms of the “forgotten symmetric functio…
View article: Chern-Dold character in complex cobordisms and theta divisors
Chern-Dold character in complex cobordisms and theta divisors Open
We show that the smooth theta divisors of general principally polarised abelian varieties can be chosen as irreducible algebraic representatives of the coefficients of the Chern-Dold character in complex cobordisms and describe the action …
View article: KdV hierarchies and quantum Novikov's equations
KdV hierarchies and quantum Novikov's equations Open
This paper begins with a review of the well-known KdV hierarchy, the $N$-th Novikov equation, and its finite hierarchy in the classical commutative case. This finite hierarchy consists of $N$ compatible integrable polynomial dynamical syst…
View article: The Mumford Dynamical System and Hyperelliptic Kleinian Functions
The Mumford Dynamical System and Hyperelliptic Kleinian Functions Open
We establish differential-algebraic theory of the Mumford dynamical system. In the framework of this theory, we introduce the $(P,Q)$-recursion, which defines a sequence of functions $P_1,P_2,\ldots$ given the first function of this sequen…
View article: Differential algebra of polytopes and inversion formulas
Differential algebra of polytopes and inversion formulas Open
We use the differential algebra of polytopes to explain the known remarkable relation of the combinatorics of the associahedra and permutohedra with the universal compositional and multiplicative inversion formulas for the formal power ser…
View article: Moduli Spaces of Weighed Pointed Stable Curves and Toric Topology of Grassmann Manifolds
Moduli Spaces of Weighed Pointed Stable Curves and Toric Topology of Grassmann Manifolds Open
View article: Todd polynomials and Hirzebruch numbers
Todd polynomials and Hirzebruch numbers Open
In 1956 Hirzebruch found an explicit formula for the denominators of the Todd polynomials, which was proved later in his joint work with Atiyah. We present a new formula for the Todd polynomials in terms of the ``forgotten symmetric functi…
View article: Random eigenvalues of graphenes and the triangulation of plane
Random eigenvalues of graphenes and the triangulation of plane Open
We analyse the numbers of closed paths of length $k\in\mathbb{N}$ on two important regular lattices: the hexagonal lattice (also called $\textit{graphene}$ in chemistry) and its dual triangular lattice. These numbers form a moment sequence…
View article: Игорь Моисеевич Кричевер (некролог)
Игорь Моисеевич Кричевер (некролог) Open
View article: The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}//(\pmb{\mathbb{C}}^{\ast})^n$ of the Grassmann manifolds $G_{n,2}$
The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}//(\pmb{\mathbb{C}}^{\ast})^n$ of the Grassmann manifolds $G_{n,2}$ Open
The complex Grassmann manifolds $G_{n,k}$ appear as one of the fundamental objects in developing an interaction between algebraic geometry and algebraic topology. The case $k=2$ is of special interest on its own as the manifolds $G_{n,2}$ …
View article: Циклические фробениусовы алгебры
Циклические фробениусовы алгебры Open
In this paper, we introduce the notion of cyclic Frobenius algebras (CF-algebras). Canonical structures of CF-algebras exist on associative and Poisson algebras. It turns out that the modern theory of integrable systems yields non-trivial …
View article: Cyclic Frobenius algebras
Cyclic Frobenius algebras Open
View article: Igor Moiseevich Krichever (obituary)
Igor Moiseevich Krichever (obituary) Open
View article: Разрешение особенностей пространств орбит $G_{n,2}/T^n$
Разрешение особенностей пространств орбит $G_{n,2}/T^n$ Open
Изучается пространство орбит $X_n = G_{n,2}/T^n$ стандартного действия компактного тора $T^n$ на комплексном многообразии Грассмана $G_{n,2}$. Описана структура множества критических точек $\operatorname {Crit}G_{n,2}$ обобщенного отображе…
View article: Cluster-permutohedra and submanifolds of flag varieties with torus actions
Cluster-permutohedra and submanifolds of flag varieties with torus actions Open
In this paper we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardo, Del R\'ıo-Francos, López-Dudet, Oliveros, and Schulte in 2010, and toric topology of manifolds of sparse isospectral Hermitian matrices.…
View article: Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions
Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions Open
The article is devoted to the classical problems about the relationships\nbetween elliptic functions and hyperelliptic functions of genus 2. It contains\nnew results, as well as a derivation from them of well-known results on these\nissues…
View article: Евгений Витальевич Щепин (к семидесятилетию со дня рождения)
Евгений Витальевич Щепин (к семидесятилетию со дня рождения) Open