Armand Borel (21 May 1923 – 11 August 2003) was a Swiss
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, born in
La Chaux-de-Fonds
La Chaux-de-Fonds (; archaic ) is a Swiss city in the canton of Neuchâtel. It is located in the Jura Mountains at an altitude of 992 metres, a few kilometres south of the French border. After Geneva, Lausanne, Biel/Bienne, and Fribourg, ...
, and was a permanent professor at the
Institute for Advanced Study
The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
in
Princeton, New Jersey
The Municipality of Princeton is a Borough (New Jersey), borough in Mercer County, New Jersey, United States. It was established on January 1, 2013, through the consolidation of the Borough of Princeton, New Jersey, Borough of Princeton and Pri ...
, 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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, in the theory of
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s, and was one of the creators of the contemporary theory of
linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...
s.
Biography
He studied at the
ETH Zürich
ETH Zurich (; ) is a public university in Zurich, Switzerland. Founded in 1854 with the stated mission to educate engineers and scientists, the university focuses primarily on science, technology, engineering, and mathematics. ETH Zurich ra ...
, 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 dynamical systems, topology and geometry.
Early life and education
Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland) ...
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
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they h ...
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 ...
s, under the influence of
Jean Leray
Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.
Life and career
He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...
and
Henri Cartan
Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology.
He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of c ...
. With
Hirzebruch
Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
, 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 to which the bundle is "twisted" and whether it possesses sections. Characterist ...
es in the early 1950s.
He collaborated with
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Early life ...
in fundamental work on
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
s, and with
Harish-Chandra
Harish-Chandra (né Harishchandra) FRS (11 October 1923 – 16 October 1983) was an Indian-American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.
Early ...
on their
arithmetic subgroups. In an algebraic group ''G'' a ''
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 subgr ...
'' ''H'' is one minimal with respect to the property that the
homogeneous space
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
''G/H'' is a
projective variety
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...
. 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
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates ...
, 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 manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a sm ...
s). 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 ...
s, and is closely related to
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open 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 prestigi ...
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 b ...
"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 (The Academy) 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 other ...
, the United States
National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
, and the
American Philosophical Society
The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...
.
He died in Princeton. He used to answer the question of whether he was related to
Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French people, French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability.
Biograp ...
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 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 a weak, algebraic notion of ...
*
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 ordi ...
*
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 subgr ...
*
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 group, ...
*
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
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algeb ...
*
Borel–de Siebenthal theory
*
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
*
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...
*
Spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathemati ...
Publications
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References
Sources
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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