Victor Matveevich Buchstaber

Russian Academy of Sciences | RAS · Steklov Mathematical Institute

We express the topological characteristics of the theta divisors and their intersections in terms of the combinatorics of permutohedra. In particular, we show that two-parameter Todd genus of theta divisor $\Theta^n$ coincides with $h$-polynomial of the corresponding permutohedron. As an application we find all the Hodge numbers of the theta diviso...

In this paper we define the parametric Korteweg-de Vries hierarchy that depends on an infinite set of graded parameters $a = (a_4,a_6,\dots)$. We show that, for any genus $g$, the Klein hyperelliptic function $\wp_{1,1}(t,\lambda)$ defined on the basis of the multidimensional sigma function $\sigma(t, \lambda)$, where $t = (t_1, t_3,\dots, t_{2g-1}...

In this paper we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardoa, Del R\'{\i}o-Francosa, L\'{o}pez-Dudeta, Oliverosa, and Schulte in 2010, and toric topology of manifolds of sparse isospectral Hermitian matrices. More precisely, we recall the notion of a cluster-permutohedron, a certain poset defined for a simp...

The paper is devoted to biography and scientific achievements of professor Victor Nikolaevich Latyshev

The paper is devoted to professor Victor Nikolaevich Latyshev

The paper is devoted to biography of professor Latyshev

В работе определена параметрическая иерархия Кортевега-де Фриза, зависящая от бесконечного набора градуированных параметров $a = (a_4,a_6,…)$. Показано, что для любого рода $g$ гиперэллиптическая функция Клейна $\wp_{1,1}(t,\lambda)$, определенная на основе многомерной сигмa-функции $\sigma(t, \lambda)$, где $t = (t_1, t_3,…, t_{2g-1})$, $\lambda =...

Получена полная классификация конечно порожденных инволютивных коммутативных двузначных групп. Построены три серии таких двузначных групп: основная, унипотентная и специальная - и показано, что любая конечно порожденная инволютивная коммутативная двузначная группа изоморфна двузначной группе, принадлежащей одной из этих серий. Получен ряд классифик...

Описаны представления групп $G_I$, $G_II$, $G_III$, $G_IV$, характеризующих симметрии пространства решений специального дважды конфлюентного уравнения Гойна. Введены категории групп, коммутант которых изоморфен группе целых чисел, и описан алгоритм категорной характеризации таких групп. Дана реализация этого алгоритма для групп $G_I,…,G_IV$. Библио...

The paper begins with a review of the well known Novikov's equations and corresponding finite KdV hierarchies. For a positive integer $N$ we give an explicit description of the $N$-th Novikov's equation and its first integrals. Its finite KdV hierarchy consists of $N$ compatible integrable polynomial dynamical systems in $\mathbb{C}^{2N}$. Then we...

In a 2004 paper by V. M. Buchstaber and D. V. Leykin, published in ``Functional Analysis and Its Applications,'' for each $g > 0$, a system of $2g$ multidimensional heat equations in a nonholonomic frame was constructed. The sigma function of the universal hyperelliptic curve of genus $g$ is a solution of this system. In the work arXiv:2007.08966 e...

The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hypere...

The focus of our paper is on the complex Grassmann manifolds $G_{n,2}$ which appear as one of the fundamental objects in developing the interaction between algebraic geometry and algebraic topology. In his well-known paper Kapranov has proved that the Deligne-Mumford compactification $\overline{\mathcal{M}}(0,n)$ of $n$-pointed curves of genus zero...

After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have been subject of much research. One part of that research is the prediction of a fullerene’s stability using topological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facets on its surface, calculations mostly were...

В работе В. М. Бухштабера и Д. В. Лейкина, опубликованной в 2004 г. в журнале «Функциональный анализ и его приложения», для каждого $g>0$ была построена система из $2g$ многомерных уравнений теплопроводности в неголономном репере. Сигма-функция универсальной гиперэллиптической кривой рода $g$ является решением этой системы. В нашей предыдущей работ...

Обзор посвящен интегрируемым полиномиальным гамильтоновым системам, ассоциированным с симметрическими степенями плоских алгебраических кривых. В центре внимания открытые авторами связи систем Штеккеля, уравнений Новикова для $g$-й стационарной иерархии Кортевега-де Фриза и координат Дубровина-Новикова на универсальном расслоении якобианов гиперэлли...

The article considers the scientific heritage of V. A. Rokhlin in algebraic topology from the point of view of the modern development of mathematics and shows the influence of his results on the development of algebraic topology up to the present. The second part of the article contains new results with fairly detailed sketches of their proofs. The...

Yang–Baxter maps (YB maps) are set-theoretical solutions to the quantum Yang–Baxter equation. For a set X = Ω × V , where V is a vector space and Ω is regarded as a space of parameters, a linear parametric YB map is a YB map Y : X × X → X × X such that Y is linear with respect to V and one has πY = π for the projection π : X × X → Ω × Ω. These cond...

В работе В. М. Бухштабера и Д. В. Лейкина, опубликованной в 2004 г. в журнале «Функциональный анализ и его приложения», для каждого $g > 0$ определена система из $2g$ многомерных уравнений Шрeдингера в магнитных полях с квадратичными потенциалами. Такие системы эквивалентны системам уравнений теплопроводности в неголономном репере. Доказано, что та...

The problem of the description of the orbit space $X_{n} = G_{n,2}/T^n$ for the standard action of the torus $T^n$ on a complex Grassmann manifold $G_{n,2}$ is widely known and it appears in diversity of mathematical questions. A point $x\in X_{n}$ is said to be a critical point if the stabilizer of its corresponding orbit is nontrivial. In this pa...

Yang--Baxter maps (YB maps) are set-theoretical solutions to the quantum Yang--Baxter equation. For a set $X=\Omega\times V$, where $V$ is a vector space and $\Omega$ is regarded as a space of parameters, a linear parametric YB map is a YB map $Y\colon X\times X\to X\times X$ such that $Y$ is linear with respect to $V$ and one has $\pi Y=\pi$ for t...

Исследуются свойства пространства $\boldsymbol{\Omega}$ решений специального дважды конфлюэнтного уравнения Гойна, тесно связанного с моделью сильношунтированного перехода Джозефсона. Описаны операторы, действующие на $\boldsymbol{\Omega}$, и соотношения в порожденной ими алгебре $\mathcal A$ над полем вещественных чисел. Структура алгебры $\mathca...

In the work by V. M. Buchstaber and D. V. Leikin for any $g > 0$ is defined a system of $2g$ multidimensional Schr\"odinger equations in magnetic fields with quadratic potentials. This systems are equivalent to systems of heat equations in nonholonomic frame. It is proved that such a system determines the sigma function of the universal hyperellipt...

We show that the theta divisors of general principally polarised abelian varieties can be chosen as smooth irreducible algebraic representatives of the coefficients of the Chern-Dold character in complex cobordisms and describe the action on them of the Landweber-Novikov operations. The link with Milnor-Hirzebruch problem about algebraic representa...

Построены алгебры Ли систем из $2g$ градуированных операторов теплопроводности $Q_0,Q_2,…,Q_{4g-2}$, определяющих сигма-функции $\sigma(z,\lambda)$ гиперэллитических кривых рода $g=1,2$ и $3$. В качестве следствия получено, что системы из трех операторов $Q_0$, $Q_2$ и $Q_4$ уже достаточно, чтобы определить сигма-функции. Оператор $Q_0$ является оп...

This survey is devoted to the classical and modern problems related to the entire function ${\sigma({\bf u};\lambda)}$, defined by a family of nonsingular algebraic curves of genus $2$, where ${\bf u} = (u_1,u_3)$ and $\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})$. It is an analogue of the Weierstrass sigma function $\sigma(u;g_2,g_3)$ o...

This survey is devoted to the classical and modern problems related to the entire function ${\sigma({\bf u};\lambda)}$, defined by a family of nonsingular algebraic curves of genus $2$, where ${\bf u} = (u_1,u_3)$ and $\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})$. It is an analogue of the Weierstrass sigma function $\sigma(u;g_2,g_3)$ o...

Integrable Systems and Algebraic Geometry - edited by Ron Donagi April 2020

In this paper we introduce a direct family of simple polytopes $P^{0}\subset P^{1}\subset\ldots$ such that for any $k$, $2\leq k\leq n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of their moment-angle manifolds $\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $\ast\subset S^{3}\hook...

In this work we give an explicit solution to the problem of differentiation of hyperelliptic functions in genus $4$ case. It is a genus $4$ analogue of the classical result of F. G. Frobenius and L. Stickelberger [F. G. Frobenius, L. Stickelberger, "Uber die Differentiation der elliptischen Functionen nach den Perioden und Invarianten", J. Reine An...

In 1896 Frobenius and Fricke had published two seemingly unrelated papers: Frobenius had started to develop his theory of $k$-characters for finite groups motivated by Dedekind's question about factorisation of the group determinant, while Fricke followed Klein's approach to the uniformization theorem. We show that in fact these two works can be na...

Professor Mikhail Ivanovich Shtogrin (born September 25, 1938) is widely known due to his contributions to discrete geometry (including regular tilings and Dirichlet-Voronoi partitions) and geometrical crystallography (including cubical complexes). The paper contains a short description of his life, scientific activities, and a photo.

In the work Lie algebras of systems of $2 g$ graded heat conduction operators $Q_0, Q_2, \ldots, Q_{4g-2}$ determining the sigma functions $\sigma(z, \lambda)$ of genus $g = 1,2$, and $3$ hyperellic curves are constructed. As a corollary, it is found that a system of three operators $Q_0, Q_2$ and $Q_4$ is already sufficient to determine the sigma...

After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have been subject of much research. One part of that research is the prediction of a fullerene's stability using topological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facets on its surface, calculations mostly were...

This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions F and G of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering W of the universal hyperelliptic curve of genus 2. Here W is the...

Conway’s topographic approach to binary quadratic forms and Markov triples is reviewed from the point of view of the theory of two-valued groups. This leads naturally to a new class of commutative two-valued groups, which we call involutive. It is shown that the two-valued group of Conway’s lax vectors plays a special role in this class. The group...

Топографический подход Конвея к бинарным квадратичным формам и тройкам Маркова рассматривается с точки зрения теории двузначных групп. Это естественно приводит к новому классу коммутативных двузначных групп, которые мы называем инволютивными. Мы показываем, что в этом классе особую роль играет двузначная группа нестрогих векторов Конвея. Группа $\m...

Статья посвящена классической задаче обращения ультраэллиптических интегралов, задаваемых базисными голоморфными дифференциалами на кривой рода 2. Базисные решения $F$ и $G$ этой задачи получены из однозначной 4-периодической мероморфной функции на абелевом накрытии $W$ универсальной гиперэллиптической кривой рода 2. В качестве $W$ мы используем не...

Professor Mikhail Ivanovich Shtogrin (born September 25, 1938) is widely known due to his contributions to discrete geometry (including regular tilings and Dirichlet-Voronoi partitions) and geometrical crystallography (including cubical complexes). The paper contains a short description of his life, scientific activities, and a photo.

В настоящей работе построена прямая последовательность $P^{0}\subset P^{1}\subset\cdots$ простых многогранников таких, что для всех $2\leq k\leq n$ в кольцах когомологий их момент-угол многообразий $\mathcal Z_{P^n}$ существуют однозначно определенные нетривиальные $k$-местные произведения Масси. Доказано, что прямая последовательность многообразий...

Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions o...

The problem of existence of nontrivial Massey products in cohomology of a space is well-known in algebraic topology and homological algebra. A number of problems in complex geometry, symplectic geometry, and algebraic topology can be stated in terms of Massey products. One of such problems is to establish formality of smooth manifolds in rational h...

We consider multi-variable sigma function of a genus $g$ hyperelliptic curve as a function of two group of variables -jacobian variables and parameters of the curve. In the theta-functional representation of sigma-function, the second group arises as periods of first and second kind differentials of the curve. We develop representation of periods i...

We show that for any face $F$ of a simple polytope $P$ the canonical equivariant homeomorphisms $h_P:\,\mathcal Z_P\to\mathcal Z_{K_P}$ and $h_F:\,\mathcal Z_F\to\mathcal Z_{K_F}$ are linked in a pentagonal commutative diagram with the maps of moment-angle manifolds and moment-angle-complexes, induced by a face embedding $i_{F,P}:\,F\to P$ and a si...

The paper is devoted to applications of functional equations to well-known problems of compact torus actions on oriented smooth manifolds. These include the problem of Hirzebruch genera of complex cobordism classes that are determined by complex, almost complex, and stably complex structures on a fixed manifold. We consider actions with connected s...

An arrow matrix is a matrix with zeros outside the main diagonal, the first row and the first column. We consider the space of Hermitian arrow -matrices with fixed simple spectrum . We prove that this space is a smooth -manifold with a locally standard torus action: we describe the topology and combinatorics of its orbit space. If , the orbit space...

In the theory of $(2n,k)$-manifolds we consider compact, closed, oriented $2n$-dimensional manifolds $M^{2n}$, with an effective action of the $k$-dimensional compact torus $T^k$, where $k\leq n$, which have the finite number of fixed points. We present the foundation axioms for this theory and prove that the structural data given by these axioms a...

We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree coho...

The canonical representation of the Klein group K4 = ℤ2⊕ℤ2 on the space ℂ* = ℂ {0} induces a representation of this group on the ring L = C[z, z⁻¹], z ∈ ℂ*, of Laurent polynomials and, as a consequence, a representation of the group K4 on the automorphism group of the group G = GL(4,L) by means of the elementwise action. The semidirect product ĜG =...

The family of complex Grassman manifolds $G_{n,k}$ with the canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogous of the moment map $\mu : G_{n,k}\to \Delta _{n,k}$ for the hypersimplex $\Delta _{n,k}$, is well known. In this paper we study the structure of the orbit space $G_{n,k}/T^n$ by developing the methods of the toric topology...

These lectures are devoted to a remarkable class of 3-dimensional polytopes, which are mathematical models of an important object in quantum physics, quantum chemistry and nanotechnology - fullerenes. The main goal is to show how results of toric topology help to build combinatorial invariants of fullerenes. Fundamental notions are introduced durin...

We describe the combinatorics of three families of simple - dimensional polytopes which play an important role in various problems of algebraic topology, hyperbolic geometry, graph theory, and their applications. The first family consists of simple polytopes with at most hexagonal faces. The second family consists of Pogorelov polytopes. The third...

We construct Lie algebras of vector fields on universal bundles $\mathcal{E}^2_{N,0}$ of symmetric squares of hyperelliptic curves of genus $g=1,2,\dots$, where $g=\left[\frac{N-1}{2}\right], \ N=3,4,\ldots$. For each of these Lie algebras, the Lie subalgebra of vertical fields has commuting generators, while the generators of the Lie subalgebra of...

A rectangular matrix is called {\it totally positive} if all its minors are positive. A point of a real Grassmanian manifold $G_{l,m}$ of $l$-dimensional subspaces in $\mathbb R^m$ is called {\it strictly totally positive} if one can normalize its Pl\"ucker coordinates to make all of them positive. Clearly if a $k\times m$-matrix, $k<m$, is totally...

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 is described in terms of the gradient of its sigma function. As an application, solutions of the corresponding families of polynomial dynamical systems in C4 with two polynomial integrals are constructed. These systems were introduced by Buchstaber and Mikhail...

We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves of genus g = 1, 2,.. For each of these Lie algebras, the Lie subalgebra of vertical fields has commuting generators, while the generators of the Lie subalgebra of projectable fields determines the canonical representation of the Lie subalgeb...

We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene $C_{20}$). A growth operation is a combinatorial opera...