Viktor Buchstaber
Encyclopedia
Viktor Matveyevich Buchstaber ' onMouseout='HidePop("15764")' href="/topics/Tashkent">Tashkent
, USSR) is a Soviet and Russia
n mathematician known for his work on algebraic topology
, homotopy
and mathematical physics
.
in K-theory
and complex cobordism theory, constructed Chern-Dold characters and the universal Todd genus in cobordism, and gave an alternative effective solution of the Milnor-Hirzebruch problem. He went on to develop a theory of double-valued formal groups that lead to the calculation of cobordism rings of complex manifold
s having symplectic coverings and to the explicit construction of what are now known as Buchstaber manifolds. He devised filtrations in Hopf algebra
s and the Buchstaber spectral sequence, which were successfully applied to the calculation of stable homotopy group
s of spheres.
He worked on the deformation theory
for mappings to groups, which lead to the solution of the Novikov problem on multiplicative subgroups in operator doubles, and to construction of the quantum group
of complex cobordism
s. He went on to treat problems related both with algebraic geometry
and integrable system
s. He is also well known for his work on sigma-functions on universal spaces of Jacobian varieties of algebraic curve
s that give effective solutions of important integrable systems. Buchstaber created an algebro-functional theory of symmetric product
s of spaces and described algebraic varieties of polysymmetric polynomials.
, he is currently a professor at the Department of Mathematics and Mechanics, Moscow State University, and is also a Professor at the School of Mathematics, University of Manchester
.
Buchstaber has an Erdős number
of 3.
In 2006 Buchstaber was elected a corresponding member of the Russian Academy of Sciences
.
Tashkent
Tashkent is the capital of Uzbekistan and of the Tashkent Province. The officially registered population of the city in 2008 was about 2.2 million. Unofficial sources estimate the actual population may be as much as 4.45 million.-Early Islamic History:...
, USSR) is a Soviet and Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
n mathematician known for his work on algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, homotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
and mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
.
Work
Buchstaber's first research work was in cobordism theory. He calculated the differential in the Atiyah-Hirzebruch spectral sequenceAtiyah-Hirzebruch spectral sequence
In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of K-theory. For a CW complex X, it relates the generalized cohomology groups...
in K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
and complex cobordism theory, constructed Chern-Dold characters and the universal Todd genus in cobordism, and gave an alternative effective solution of the Milnor-Hirzebruch problem. He went on to develop a theory of double-valued formal groups that lead to the calculation of cobordism rings of complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s having symplectic coverings and to the explicit construction of what are now known as Buchstaber manifolds. He devised filtrations in Hopf algebra
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
s and the Buchstaber spectral sequence, which were successfully applied to the calculation of stable homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s of spheres.
He worked on the deformation theory
Deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach...
for mappings to groups, which lead to the solution of the Novikov problem on multiplicative subgroups in operator doubles, and to construction of the quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
of complex cobordism
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
s. He went on to treat problems related both with algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and integrable system
Integrable system
In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...
s. He is also well known for his work on sigma-functions on universal spaces of Jacobian varieties of algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s that give effective solutions of important integrable systems. Buchstaber created an algebro-functional theory of symmetric product
Symmetric product
In mathematics, the term symmetric product can refer to:* The symmetric part of a tensor* The symmetric product of an algebraic curve* The infinite symmetric product SP of a space X in algebraic topology...
s of spaces and described algebraic varieties of polysymmetric polynomials.
Academic career
Buchstaber gained his Ph.D. in 1970 under Novikov and Dr Sci in 1984 from Moscow State UniversityMoscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
, he is currently a professor at the Department of Mathematics and Mechanics, Moscow State University, and is also a Professor at the School of Mathematics, University of Manchester
School of Mathematics, University of Manchester
The School of Mathematics at the University of Manchester is one of the largest mathematics departments in the United Kingdom, with around 80 academic staff and an undergraduate intake of roughly 400 a year and another 200 postgraduate students...
.
Buchstaber has an Erdős number
Erdos number
The Erdős number describes the "collaborative distance" between a person and mathematician Paul Erdős, as measured by authorship of mathematical papers.The same principle has been proposed for other eminent persons in other fields.- Overview :...
of 3.
In 2006 Buchstaber was elected a corresponding member of the Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....
.
External links
- Home page at Russian Academy of Sciences http://www.mi.ras.ru/~buchstab/index.html
- Birthday tribute in Moscow Mathematical Journal http://www.ams.org/distribution/mmj/vol3-1-2003/buchstaber-60.html