• Source: Armand Borel

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.




Biography


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. With Hirzebruch, he significantly developed the theory of characteristic classes in the early 1950s.
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 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 United States National Academy of Sciences, and the American Philosophical Society.
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
Borel cohomology
Borel conjecture
Borel construction
Borel subgroup
Borel subalgebra
Borel fixed-point theorem
Borel's theorem
Borel–de Siebenthal theory
Borel–Moore homology
Baily–Borel compactification
Linear algebraic group
Spin structure




Publications


Borel, Armand (1960), Seminar on transformation groups, With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. Annals of Mathematics Studies, No. 46, Princeton University Press, MR 0116341
Borel, Armand (1964) [1957], Cohomologie des espaces localement compacts d'après J. Leray. Exposés faits au séminaire de Topologie algébrique de l'École Polytechnique Fédérale, printemps 1951, Lecture Notes in Mathematics (in French), vol. 2 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/BFb0097851, ISBN 978-3-540-03179-6, MR 0174045
Borel, Armand (1967) [1954], Halpern, Edward (ed.), Topics in the homology theory of fibre bundles, Lecture Notes in Mathematics, vol. 36, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0096867, ISBN 978-3-540-03907-5, MR 0221507
Borel, Armand (1969), Introduction aux groupes arithmétiques, Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341 (in French), Paris: Hermann, MR 0244260
Borel, Armand (1972), Représentations de groupes localement compacts, Lecture Notes in Mathematics, vol. 276, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058407, ISBN 978-3-540-05926-4, MR 0414779
Borel, Armand (1991) [1969], Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012
Borel, Armand (2008) [1984], Intersection cohomology, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4764-3, MR 0788171
Borel, Armand; Grivel, Pierre-Paul; Kaup, Burchard; Haefliger, André; Malgrange, Bernard; Ehlers, Fritz (1987), Algebraic D-modules, Perspectives in Mathematics, vol. 2, Boston, MA: Academic Press, ISBN 978-0-12-117740-9, MR 0882000
Borel, Armand (1997), Automorphic forms on SL2(R), Cambridge Tracts in Mathematics, vol. 130, Cambridge University Press, ISBN 978-0-521-58049-6, MR 1482800
Borel, Armand (1998), Semisimple groups and Riemannian symmetric spaces, Texts and Readings in Mathematics, vol. 16, New Delhi: Hindustan Book Agency, ISBN 978-81-85931-18-0, MR 1661166
Borel, Armand; Wallach, Nolan (2000) [1980], Continuous cohomology, discrete subgroups, and representations of reductive groups, Mathematical Surveys and Monographs, vol. 67 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0851-1, MR 1721403
Borel, Armand (2001), Essays in the History of Lie Groups and Algebraic Groups, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0288-5, MR 1847105
Borel, Armand (1983), Œuvres: collected papers, vol. I, II, III, Berlin, New York: Springer-Verlag, ISBN 978-3-540-12126-8, MR 0725852
Borel, Armand (2001), Œuvres: collected papers, vol. IV, Berlin, New York: Springer-Verlag, ISBN 978-3-540-67640-9, MR 1829820
Borel, Armand; Ji, Lizhen (2006), Compactifications of symmetric and locally symmetric spaces, Mathematics: Theory & Applications, Boston, MA: Birkhäuser Boston, doi:10.1007/0-8176-4466-0, ISBN 978-0-8176-3247-2, MR 2189882




References






Sources


"Special issue dedicated to the memory of Professor Armand Borel, 1923–2003", Asian Journal of Mathematics, 8 (4), 2004
Arthur, James; Bombieri, Enrico; Chandrasekharan, Komaravolu; Hirzebruch, Friedrich; Prasad, Gopal; Serre, Jean-Pierre; Springer, Tonny A.; Tits, Jacques (2004), "Armand Borel (1923--2003)", Notices of the American Mathematical Society, 51 (5): 498–524, ISSN 0002-9920, MR 2046057
Haefliger, André (2004), "Armand Borel (1923--2003)", Gazette des Mathématiciens (102): 7–14, ISSN 0224-8999, MR 2108056
O'Connor, John J.; Robertson, Edmund F., "Armand Borel", MacTutor History of Mathematics Archive, University of St Andrews
Springer, Tonny A. (2007), "Armand Borel's work in the theory of linear algebraic groups", Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., Mumbai: Tata Inst. Fund. Res., pp. 1–11, MR 2348899




External links


"Armand Borel" Archived 2015-11-21 at the Wayback Machine – obituary on Institute for Advanced Study website
Armand Borel at the Mathematics Genealogy Project
Mark Goresky, "Armand Borel", Biographical Memoirs of the National Academy of Sciences (2019)

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• Armand Borel
• Borel–Weil–Bott theorem
• Borel
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• Émile Borel
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• Borel–Moore homology
• Borel fixed-point theorem
• Armand (name)
• Borel's theorem