Swiss mathematician (1923–2003)
Armand Borel (21 May 1923 – 11 August 2003) was a Swissmathematician , born inLa Chaux-de-Fonds , and was a permanent professor at theInstitute for Advanced Study inPrinceton, New Jersey , United States from 1957 to 1993. He worked inalgebraic topology , in the theory ofLie groups , and was one of the creators of the contemporary theory oflinear algebraic groups .
He studied at theETH Zürich , where he came under the influence of the topologistHeinz Hopf and Lie-group theoristEduard Stiefel . He was in Paris from 1949: he applied the Lerayspectral sequence to the topology of Lie groups and theirclassifying spaces , under the influence ofJean Leray andHenri Cartan . WithHirzebruch , he significantly developed the theory ofcharacteristic classes in the early 1950s.
He collaborated withJacques Tits in fundamental work onalgebraic groups , and withHarish-Chandra on theirarithmetic subgroups . In an algebraic groupG aBorel subgroup H is one minimal with respect to the property that thehomogeneous space G/H is aprojective variety . For example, ifG is GLn H to be the subgroup of upper triangular matrices. In this case it turns out thatH is a maximalsolvable subgroup , and that theparabolic subgroups P betweenH andG have a combinatorial structure (in this case the homogeneous spacesG/P are the variousflag manifolds ). Both those aspects generalize, and play a central role in the theory.
TheBorel−Moore homology theory applies to generallocally compact spaces , and is closely related tosheaf theory.
He published a number of books, including a work on the history of Lie groups. In 1978 he received theBrouwer Medal [ 1] 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 theAmerican Academy of Arts and Sciences , the United StatesNational Academy of Sciences , and theAmerican Philosophical Society .[ 2] [ 3] [ 4]
He died in Princeton. He used to answer the question of whether he was related toÉmile Borel alternately by saying he was a nephew, and no relation.
"I feel that what mathematics needs least are pundits who issue prescriptions or guidelines for presumably less enlightened mortals." (Oeuvres IV, p. 452)
Borel, Armand (1960),Seminar on transformation groups G. Bredon ,E. E. Floyd ,D. Montgomery ,R. Palais . Annals of Mathematics Studies, No. 46,Princeton University Press ,MR 0116341 [ 5] 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 ,MR 0174045 ISBN 978-3-540-03179-6 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 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 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 [ 6] Borel, Armand (1998),Semisimple groups and Riemannian symmetric spaces ISBN 978-81-85931-18-0 MR 1661166 Borel, Armand ;Wallach, Nolan (2000) [1980],Continuous cohomology, discrete subgroups, and representations of reductive groups 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 [ 7] 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 "Special issue dedicated to the memory of Professor Armand Borel, 1923–2003" ,Asian Journal of Mathematics ,8 (4), 2004Arthur, 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
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