Armand Borel (nonfiction): Difference between revisions
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He studied at the ETH Zürich, where he came under the influence of the topologist [[Heinz Hopf (nonfiction)|Heinz Hopf]] and Lie-group theorist [[Eduard Stiefel (nonfiction)|Eduard Stiefel]]. He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their classifying spaces, under the influence of [[Jean Leray (nonfiction)|Jean Leray]] and [[Henri Cartan (nonfiction)|Henri Cartan]]. | He studied at the ETH Zürich, where he came under the influence of the topologist [[Heinz Hopf (nonfiction)|Heinz Hopf]] and Lie-group theorist [[Eduard Stiefel (nonfiction)|Eduard Stiefel]]. He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their classifying spaces, under the influence of [[Jean Leray (nonfiction)|Jean Leray]] and [[Henri Cartan (nonfiction)|Henri Cartan]]. | ||
He collaborated with [[Jacques Tits (nonfiction)|Jacques Tits]] in fundamental work on algebraic groups, and with [[Harish-Chandra (nonfiction)|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 G/H is a projective variety. 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 homogeneous spaces G/P are the various flag manifolds). Both those aspects generalize, and play a central role in the theory. | He collaborated with [[Jacques Tits (nonfiction)|Jacques Tits]] in fundamental work on algebraic groups, and with [[Harish-Chandra (nonfiction)|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 ''G''/''H'' is a projective variety. 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 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 spaces, and is closely related to sheaf theory. | The Borel−Moore homology theory applies to general locally compact spaces, and is closely related to sheaf theory. | ||
Revision as of 11:29, 26 December 2017
Armand Borel (21 May 1923 – 11 August 2003) 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 groups, and was one of the creators of the contemporary theory of linear algebraic groups.
He studied at the ETH Zürich, where he came under the influence of the topologist Heinz Hopf and 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 spaces, under the influence of Jean Leray and Henri Cartan.
He collaborated with Jacques Tits in fundamental work on algebraic groups, 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 G/H is a projective variety. 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 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 spaces, and is closely related to sheaf theory.
He published a number of books, including a work on the history of Lie groups. In 1978 he received the Brouwer Medal and 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 favor of high quality in mathematical research and the propagation of new ideas" (motivation of the Balzan General Prize Committee).
He used to answer the question of whether he was related to Émile Borel alternately by saying he was a nephew, and no relation.
He died in Princeton.
In the News
Fiction cross-reference
Nonfiction cross-reference
- Algebraic topology (nonfiction)
- Émile Borel (nonfiction)
- Henri Cartan (nonfiction)
- Harish-Chandra (nonfiction)
- Heinz Hopf (nonfiction)
- Jean Leray (nonfiction)
- Lie group (nonfiction)
- Mathematician (nonfiction)
- Eduard Stiefel (nonfiction)
- Jacques Tits (nonfiction)
External links:
- Armand Borel @ Wikipedia