Ernest Borissowitsch Winberg

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Ernest Borissowitsch Winberg ( Russian Эрнест Борисович Винберг , English transcription Ernest Borisovich Vinberg; born July 26, 1937 in Moscow ; † May 12, 2020 ibid) was a Russian mathematician who dealt with algebra , especially the representation theory of groups , and geometry .

Life

Winberg graduated from Lomonosov University in 1959 and did his doctorate there (Russian candidate) in 1962 under Eugene Dynkin and Ilya Pyatetskij-Shapiro . He taught from 1961 at the Chair of Algebra at Lomonosov University (from 1966 with an assistant professorship, from 1991 with full professorship) and was also a professor at the Independent University in Moscow.

He was on the governing committee of the Moscow Mathematical Society. Vinberg received the Humboldt Research Award . In 1983 he was invited speaker at the International Congress of Mathematicians in Warsaw (Discrete reflection groups in Lobachevsky spaces) . In 2010 he was elected to the American Academy of Arts and Sciences .

Victor Kac (1968) and Boris Weisfeiler (1970) were among his doctoral students .

He died on May 12, 2020 at the age of 82.

plant

In his first work, Vinberg classified homogeneous spaces of Lie groups with invariant linear forms of connection . He then dealt with homogeneous convex cones , which he classified in the non-self-dual case (via connections to Jordan algebras ), where he discovered a new class of non-associative algebras named after him (Vinberg algebras). He also gave the first example of a non- self-dual homogeneous convex cone. In the 1980s he studied invariant cones in Lie algebras .

From the 1960s he began a systematic study of discrete crystallographic reflection groups . In 1983 he proved that there are no cocompact hyperbolic reflection groups in hyperbolic spaces of thirty or more dimensions . He also examined the arithmeticity of hyperbolic reflection groups and proved that there are no discrete arithmetic reflection groups of the non-compact type in hyperbolic spaces of thirty and more dimensions.

In the invariant theory , for example, in 1976 with Victor Kac and VL Popov (who received his doctorate in 1972), he classified the simple, connected irreducible linear algebraic groups with a free algebra of invariants.

In the 1970s he promoted the study of locally transitive effects of algebraic groups in algebraic varieties and equivariant embeddings of homogeneous spaces. This led, among other things, to a theory of the representation of toric varieties in convex geometry by rational polyhedra and fans . In 1986 he introduced a measure of complexity for the effects of a reductive group in irreducible algebraic varieties.

In the 2000s he studied commutative homogeneous spaces of Lie groups, that is, such homogeneous spaces with a commutative algebra of invariant differential operators .

Named after him are the Vinberg algorithm for the construction of fundamental domains and the Koecher-Vinberg theorem about the correspondence between formally real Jordan algebras and symmetrical convex cones.

Fonts

  • Linear representations of groups . Birkhäuser, 1989
  • A Course in Algebra . American Mathematical Society (AMS), 2003
  • as editor and co-author: Lie groups and invariant theory . AMS, 2005 (from Vinberg therein among others: Construction of the exceptional simple Lie algebras)
  • with AL Onishchik : Lie groups and algebraic groups . Springer, 1990
  • with VV Gorbatsevich, AL Onishchik: Foundations of Lie groups and Lie transformation groups . Springer, 1997
  • Hyperbolic reflection groups . Russian Mathematical Surveys, Vol. 40, 1985, pp. 31-75
  • as editor: Geometry II . Encyclopedia of Mathematical Sciences, Springer 1991, (therein Vinberg and others: Geometry of spaces of constant curvature , Discrete groups of motions of spaces of constant curvature )

Web links

Individual evidence

  1. ↑ He also wrote the article with Onishchik and Karpelevich on Dynkin's work on Lie groups in his Selected Papers , which were published in 1999 by the American Mathematical Society by Juschkewitsch, Seitz, Onishchik.
  2. ↑ Кирилл Владимирович Семенов (Kirill Vladimirovich Semenov): Скончался Э.Б.Винберг. In: math.msu.ru. May 12, 2020, accessed May 12, 2020 (Russian).
personal data
SURNAME Winberg, Ernest Borissowitsch
ALTERNATIVE NAMES Винберг, Эрнест Борисович (Russian spelling); Vinberg, Ernest Borisovich (English transcription)
BRIEF DESCRIPTION Russian mathematician
DATE OF BIRTH July 26, 1937
PLACE OF BIRTH Moscow
DATE OF DEATH May 12, 2020
Place of death Moscow