Victor Matveevich Buchstaber
Russian Academy of Sciences
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Featured researches published by Victor Matveevich Buchstaber.
International Mathematics Research Notices | 2001
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
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
Victor Matveevich Buchstaber; V. Z. Enolskii; D. V. Leykin
A_{n-1}
Siam Journal on Mathematical Analysis | 1997
Harry Braden; Victor Matveevich Buchstaber
case, we discuss the relation with elliptic Calogero-Moser integrable
Functional Analysis and Its Applications | 2000
Victor Matveevich Buchstaber; V. Z. Enolskii; D. V. Leykin
n
Proceedings of the Steklov Institute of Mathematics | 2008
Victor Matveevich Buchstaber
-body problems, and discuss the quantization (
Transformation Groups | 1997
Victor Matveevich Buchstaber; E. G. Rees
q
Functional Analysis and Its Applications | 2002
Victor Matveevich Buchstaber; D. V. Leykin
-analogue) of our construction.
Archive | 2012
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
Victor Matveevich Buchstaber; E. G. Rees
The general analytic solution to the functional equation