Armand Borel
Encyclopedia
Armand Borel was a Swiss
mathematician
, born in La Chaux-de-Fonds
, 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
, in the theory of Lie group
s, and was one of the creators of the contemporary theory of linear algebraic group
s.
. He came under the influence of the topologist Heinz Hopf
, and the Lie group theorist Eduard Stiefel
. He was in Paris from 1949: he applied the Leray spectral sequence
to the topology of Lie groups and their classifying space
s, under the influence of Jean Leray
and Henri Cartan
.
He collaborated with Jacques Tits
in fundamental work on algebraic group
s, and with Harish-Chandra
on their arithmetic subgroups
. In an algebraic group G a Borel subgroup H is one such that the homogeneous space
G/H is a projective variety, and as small as possible. For example if G is GLn 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 homogenous spaces G/P are the various flag manifold
s). Both those aspects generalize, and play a central role in the theory.
The Borel−Moore homology theory applies to general locally compact space
s, and is closely related to sheaf
theory.
He published a number of books, including a work on the history of Lie groups. In 1992 he was awarded the Balzan Prize
"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 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.
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, born in La Chaux-de-Fonds
La Chaux-de-Fonds
La Chaux-de-Fonds is a Swiss city of the district of La Chaux-de-Fonds in the canton of Neuchâtel. It is located in the Jura mountains at an altitude of 1000 m, a few kilometres from the French border. After Geneva and Lausanne, it is the third largest city of Romandie, the French-speaking part of...
, and was a permanent professor at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
, United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
from 1957 to 1993. He worked in 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...
, in the theory of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, and was one of the creators of the contemporary theory of linear algebraic group
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
s.
Biography
He studied at the ETH ZürichETH Zurich
The Swiss Federal Institute of Technology Zurich or ETH Zürich is an engineering, science, technology, mathematics and management university in the City of Zurich, Switzerland....
. He came under the influence of the topologist Heinz Hopf
Heinz Hopf
Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...
, and the Lie group theorist Eduard Stiefel
Eduard Stiefel
Eduard L. Stiefel 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 the Stiefel–Whitney classes of a real vector bundle, thus co-founding the study of...
. He was in Paris from 1949: he applied the Leray spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
to the topology of Lie groups and their classifying space
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 by a free action of G...
s, under the influence of Jean Leray
Jean Leray
Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....
and Henri Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
.
He collaborated with Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
in fundamental work on algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s, and with Harish-Chandra
Harish-Chandra
Harish-Chandra was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups. -Life:...
on their arithmetic subgroups
Arithmetic group
In mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...
. In an algebraic group G a Borel subgroup H is one such that the homogeneous space
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 continuously by symmetry in a transitive way. A special case of this is when the topological group,...
G/H is a projective variety, and as small as possible. For example if G is GLn 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
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
, and that the parabolic subgroups
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 group GLn ,...
P between H and G have a combinatorial structure (in this case the homogenous spaces G/P are the various flag manifold
Flag manifold
In mathematics, a generalized 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 smooth or complex manifold, called a real or complex flag manifold...
s). Both those aspects generalize, and play a central role in the theory.
The Borel−Moore homology theory applies to general locally compact space
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.-Formal definition:...
s, and is closely related to sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
theory.
He published a number of books, including a work on the history of Lie groups. In 1992 he was awarded the Balzan Prize
Balzan Prize
The International Balzan Prize Foundation awards four annual monetary prizes to people or organisations who have made outstanding achievements in the fields of humanities, natural sciences, culture, as well as for endeavours for peace and the brotherhood of man.-Rewards and assets:Each year the...
"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 died in Princeton. He used to answer the question of whether he was related to Émile Borel
Émile Borel
Félix Édouard Justin Émile Borel was a French mathematician and politician.Borel was born in Saint-Affrique, Aveyron. Along with René-Louis Baire and Henri Lebesgue, he was among the pioneers of measure theory and its application to probability theory. The concept of a Borel set is named in his...
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)External links
- "Armand Borel" - obituary on Institute for Advanced Study website