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Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in
La Chaux-de-Fonds La Chaux-de-Fonds () is a Swiss city in the canton of Neuchâtel. It is located in the Jura mountains at an altitude of 1000 m, a few kilometers south of the French border. After Geneva, Lausanne and Fribourg, it is the fourth largest city ...
, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in
algebraic topology Algebraic topology is a branch of mathematics that 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 ...
, in the theory of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s, and was one of the creators of the contemporary theory of linear algebraic groups.


Biography

He studied at the ETH Zürich, where he came under the influence of the topologist
Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Eliza ...
and Lie-group theorist
Eduard Stiefel Eduard L. Stiefel (21 April 1909 – 25 November 1978) was a Swiss mathematician. Together with Cornelius Lanczos and Magnus Hestenes, he invented the conjugate gradient method, and gave what is now understood to be a partial construction of th ...
. He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
s, under the influence of Jean Leray and Henri Cartan. With Hirzebruch, he significantly developed the theory of
characteristic class In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
es in the early 1950s. He collaborated with Jacques Tits in fundamental work on
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. ...
s, and with Harish-Chandra on their arithmetic subgroups. In an algebraic group ''G'' a ''Borel subgroup'' ''H'' is one minimal with respect to the property that the
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of '' ...
''G/H'' is a projective variety. For example, if ''G'' is GL''n'' then we can take ''H'' to be the subgroup of upper triangular matrices. In this case it turns out that H is a maximal solvable subgroup, and that the parabolic subgroups ''P'' between ''H'' and ''G'' have a combinatorial structure (in this case the homogeneous spaces ''G/P'' are the various flag manifolds). Both those aspects generalize, and play a central role in the theory. The Borel−Moore homology theory applies to general
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
s, and is closely related to
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * '' The Sheaf'', a student-run newspaper s ...
theory. He published a number of books, including a work on the history of Lie groups. In 1978 he received the
Brouwer Medal The Brouwer Medal is a triennial award presented by the Royal Dutch Mathematical Society and the Royal Netherlands Academy of Sciences. The Brouwer Metal gets its name from Dutch mathematician L. E. J. Brouwer and is the Netherlands’ most presti ...
and in 1992 he was awarded the
Balzan Prize The International Balzan Prize Foundation awards four annual monetary prizes to people or organizations who have made outstanding achievements in the fields of humanities, natural sciences, culture, as well as for endeavours for peace and the br ...
"For his fundamental contributions to the theory of Lie groups, algebraic groups and arithmetic groups, and for his indefatigable action in favour of high quality in mathematical research and the propagation of new ideas" (motivation of the Balzan General Prize Committee). He was a member of the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...
, the United States National Academy of Sciences, and the
American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
. He died in Princeton. He used to answer the question of whether he was related to Émile Borel alternately by saying he was a nephew, and no relation.


Famous quotations

"I feel that what mathematics needs least are pundits who issue prescriptions or guidelines for presumably less enlightened mortals." (Oeuvres IV, p. 452)


See also

*
Borel–Weil–Bott theorem In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, ...
* Borel cohomology *
Borel conjecture In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that ...
*
Borel construction In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ord ...
*
Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
*
Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie grou ...
* Borel fixed-point theorem *
Borel's theorem In topology, a branch of mathematics, Borel's theorem, due to , says the cohomology ring of a classifying space or a classifying stack is a polynomial ring. See also *Atiyah–Bott formula In algebraic geometry, the Atiyah–Bott formula say ...
*
Borel–de Siebenthal theory In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have ''maximal rank'', i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jean de Siebenthal ...
*
Borel–Moore homology In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960. For reasonable compact spaces, Borel−Moore homology coincides with the usual ...
*
Baily–Borel compactification In mathematics, the Baily–Borel compactification is a compactification of a quotient of a Hermitian symmetric space by an arithmetic group, introduced by . Example *If ''C'' is the quotient of the upper half plane by a congruence subgroup In ...
* Linear algebraic group * Spin structure


Publications

* * * * * * * * * * * * * * *


References


Sources

* * * * *


External links


"Armand Borel"
– obituary on Institute for Advanced Study website *
Mark Goresky, "Armand Borel", Biographical Memoirs of the National Academy of Sciences (2019)
{{DEFAULTSORT:Borel, Armand 1923 births 2003 deaths ETH Zurich alumni Institute for Advanced Study faculty Topologists Algebraic geometers Brouwer Medalists 20th-century Swiss mathematicians Nicolas Bourbaki Members of the French Academy of Sciences Members of the United States National Academy of Sciences Group theorists People from La Chaux-de-Fonds Members of the American Philosophical Society