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Dive into the research topics where Victor Matveevich Buchstaber is active.

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Featured researches published by Victor Matveevich Buchstaber.


International Mathematics Research Notices | 2001

Tangential structures on toric manifolds, and connected sums of polytopes

Victor Matveevich Buchstaber; Nigel Ray

We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples Bi,j, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the Bi,j allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruchs famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum # for simple n-dimensional polytopes; when Pn is a product of simplices, we describe Pn#Qn by applying an appropriate sequence of pruning operators, or hyperplane cuts, to Qn.


Duke Mathematical Journal | 1994

Elliptic Dunkl operators, root systems, and functional equations

Victor Matveevich Buchstaber; Giovanni Felder; A. V. Veselov

We consider generalizations of Dunkls differential-difference operators associated with groups generated by reflections. The commutativity condition is equivalent to certain functional equations. These equations are solved in many cases. In particular, solutions associated with elliptic curves are constructed. In the


Functional Analysis and Its Applications | 1999

Rational Analogs of Abelian Functions

Victor Matveevich Buchstaber; V. Z. Enolskii; D. V. Leykin

A_{n-1}


Siam Journal on Mathematical Analysis | 1997

The general analytic solution of a functional equation of addition type

Harry Braden; Victor Matveevich Buchstaber

case, we discuss the relation with elliptic Calogero-Moser integrable


Functional Analysis and Its Applications | 2000

Uniformization of jacobi varieties of trigonal curves and nonlinear differential equations

Victor Matveevich Buchstaber; V. Z. Enolskii; D. V. Leykin

n


Proceedings of the Steklov Institute of Mathematics | 2008

Ring of Simple Polytopes and Differential Equations

Victor Matveevich Buchstaber

-body problems, and discuss the quantization (


Transformation Groups | 1997

Multivalued groups, their representations and Hopf algebras

Victor Matveevich Buchstaber; E. G. Rees

q


Functional Analysis and Its Applications | 2002

Polynomial Lie Algebras

Victor Matveevich Buchstaber; D. V. Leykin

-analogue) of our construction.


Archive | 2012

Combinatorial 2-truncated Cubes and Applications

Victor Matveevich Buchstaber; Vadim Dmitrievich Volodin

In the theory of Abelian functions on Jacobians, the key role is played by entire functions that satisfy the Riemann vanishing theorem (see, for instance, [9]). Here we introduce polynomials that satisfy an analog of this theorem and show that these polynomials are completely characterized by this property. By rational aalalogs of Abel ian functions we mean logarithmic derivatives of orders /> 2 of tlmse polynomials. We call the polynomials thus obtained the Schur-Weierstrass polynomials because they are constructed from classical Schur polynomials, which, however, correspond to special partitions related to Weierstrass sequences. Recently, in connection with the problem of constructing rational solutions of nonlinear integrable equat ions [1, 7], special attention was focused on Schur polynomials [5, 6]. Since a Schur polynomial corresponding to all arbitrary partition leads to a rational solution of the Kadomtsev-Petviashvili hierarchy, tile problem of connecting the above solutions with those defined in terms of Abelian functions on Jacobians naturally arose. Our results open the way toward solving this problem on the basis of the Riemann vanishing theorem. We demons t ra t e our approach by the example of Weierstrass sequences defined by a pair of coprime numbers n and s. Each of these sequences generates a class of plane curves of genus g ---(n 1)(s 1)/2 defined by equat ions of the form


Archive | 1998

Multivalued Groups, n-Hopf Algebras and n-Ring Homomorphisms

Victor Matveevich Buchstaber; E. G. Rees

The general analytic solution to the functional equation

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Taras Panov

Moscow State University

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E. G. Rees

University of Edinburgh

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E. Yu. Bunkova

Russian Academy of Sciences

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Nigel Ray

University of Manchester

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