Armand Borel

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Armand Borel in 1967

Armand Borel (born May 21, 1923 in La Chaux-de-Fonds , Switzerland ; † August 11, 2003 in Princeton , USA ) was a Swiss mathematician.

Life

Borel attended school in Geneva as well as several private schools. From 1942 he studied mathematics and physics at the ETH Zurich , especially with the topologists Heinz Hopf and Eduard Stiefel , with the diploma under Stiefel in 1947. His studies were interrupted by military service. From 1947 to 1949 he was an assistant at the ETH. In 1949/50 he was in Paris with Henri Cartan and Jean Leray on a CNRS scholarship. There he also got to know the members of the Bourbaki circle and their students ( Jean Dieudonné , Laurent Schwartz , Roger Godement , Pierre Samuel , Jacques Dixmier ) and befriended many of them, especially Jean-Pierre Serre . Soon afterwards he became a member of Bourbaki himself. Leray became Borel's doctoral supervisor (dissertation 1952 in Paris: Sur La Cohomologie des Espaces Fibers Principaux et des Espaces Homogènes de Groupes de Lie Compacts ). Between 1950 and 1952 he was a deputy chair in Geneva and gave lectures at the ETH Zurich, which led to a book on Leray's ideas in topology ( Cohomologie des espaces localement compacts, d'après J Leray ). From Geneva and Zurich he traveled frequently to Paris. In 1952 he married Gabrielle ("Gaby") Aline Pittet, with whom he had two daughters.

From 1952 to 1954 he was at the Institute for Advanced Study in Princeton , where he a. a. worked with Friedrich Hirzebruch . In 1954 he studied at the University of Chicago with André Weil , from whom he mainly learned algebraic geometry and number theory, and from 1955 to 1957 he worked as a professor at the ETH Zurich. From 1957 to 1993 he was a professor at the Institute for Advanced Study at Princeton. He was also a professor at the ETH 1983-1986 and together with Jürgen Moser 1984-1986 director of the Research Institute of Mathematics and had also numerous guest professorships, z. B. in India at the Tata Institute of Fundamental Research in Bombay (1961, 1983, 1990) and in Hong Kong 1999 to 2001. He traveled extensively and had residences both in Princeton and on Lake Geneva.

plant

He initially dealt with the topology of Lie groups in Zurich and Paris . He turned the spectral sequences of Jean Leray on the topology of Lie groups and their classifying spaces ( "classifying spaces"). These spaces classify fiber bundles (in physics gauge theories) with Lie groups G as structural groups. The cohomology groups of these spaces provide the characteristic classes , e.g. B. in the case of unitary groups the Chern classes .

He was (with Serre) the main author of the volume on Lie groups and Lie algebras by Bourbaki (published in several parts from 1960). This book differs significantly in its wealth of “concrete” details from the other, mostly very abstract Bourbaki volumes.

In addition to his work in algebraic topology and the theory of Lie groups, he dealt with algebraic groups . a. collaborated with Jacques Tits , and with arithmetic groups (including collaboration with Harish-Chandra ). His work on algebraic groups in the mid-1950s changed the whole field and enabled Claude Chevalley to classify semi-simple groups over any algebraically closed fields . With Friedrich Hirzebruch in the case of the unitary group and in general with André Weil he showed that Hermann Weyl's character formulas for the irreducible representations of connected compact Lie groups G result from the Hirzebruch-Riemann-Roch theorem , applied to the (algebraic ) Group of quotients G / T (T = maximum torus of G), which is the fiber in the fiber bundle of the associated classifying spaces of G and T. The Weyl group of the Lie algebra (interchange group of the roots) operates on the fibers , which in the case of the unitary group is the symmetrical group, with an associated decomposition of the fiber into plume manifolds. The notified by Borel Borel subgroup H of an algebraic group is defined in that the homogeneous space G / H is projective and as "small" as possible. Example: G = general linear group GL (n) H = Area of upper triangular matrices, where H is a maximum resolvable sub-group , and the "parabolic groups" P between H and G, the flag manifolds form (flag manifolds). He also wrote Borel's tightness theorem .

At the same time, Hirzebruch and Borel proved in their work from 1958 that an orientable fiber bundle defines a spin structure on a manifold if and only if the second Stiefel-Whitney class of the bundle vanishes.

In the field of group theory and its application in number theory (e.g. in the sense of the Langlands program ) he also worked with Jean-Pierre Serre . With this he also wrote an essay in which Grothendieck's generalization of the Riemann-Roch theorem was first published.

In a paper published in 1974 he calculated the algebraic K-theory of number fields and their wholeness rings (except for torsion). The Borel regulator in the K theory of number fields is named after him .

Borel-Moore homology is a homology theory for locally compact spaces , in which every (not necessarily compact) orientable manifold has a fundamental class.

Equivariate homology is also occasionally referred to as Borel homology.

The Baily-Borel compactification in the theory of algebraic geometry is named after him and Walter Baily . In relation to special arithmetic groups, it makes symmetrical quotient spaces compact (closed, completed) and can be represented with modular forms.

Various conjectures are named after Borel, for example the Borel conjecture in the topology. It arose from a question he asked Serre in 1953, stating that closed manifolds whose higher homotopy groups vanish ( aspherical manifolds) and whose fundamental groups are isomorphic are homeomorphic . The assumption is open. Another Borel conjecture concerns the computation of the complex cohomology of arithmetic groups, which according to the conjecture is given by special automorphic functions. It was proven by Jens Franke .

Honors and memberships

In 1992 he received the Balzan Prize . In 1991 he received the Leroy P. Steele Prize from the American Mathematical Society . In 1962 he gave a plenary lecture at the International Congress of Mathematicians in Stockholm ( Arithmetic Properties of Linear Algebraic Groups ) and in 1974 he was invited speaker at the ICM in Vancouver ( Cohomology of arithmetic groups ). In 1978 he received the Brouwer Medal . He was a member of the American Academy of Arts and Sciences (since 1977), the Académie des sciences (since 1981), the American Philosophical Society (since 1985) and the National Academy of Sciences (since 1987).

Others

Borel was very interested in music and organized a. a. Concerts of Indian and jazz music.

Fonts

Books:

  • Oeuvres (Collected Papers) , 4 volumes, Springer 1983 to 2001
  • Topics in the homology theory of fiber bundles , Springer, Lecture notes in mathematics, 1967 (Chicago Lectures from 1954)
  • Linear algebraic groups , New York, Benjamin 1969, Springer 1991
  • Automorphic forms on SL 2 (R) , Cambridge University Press 1997
  • Semisimple groups and Riemannian symmetric spaces , Delhi: Hindustan Book Agency 1998
  • Editor (with Bill Casselman) and co-author: Automorphic forms, representations and L-functions , 2 vol., AMS symposium in pure mathematics 1979, online here: Automorphic Forms, Representations, and L-Functions / pspum31 and here: Automorphic Forms, Representations , and L-Functions /pspum33.2
  • Editor (with Mostow) and co-author: Algebraic groups and discontinuous subgroups , AMS 1966 (Symposium in Pure Mathematics, Boulder / Colorado 1965), online here: Algebraic Groups and Discontinuous Subgroups / pspum9
  • Editor and co-author of the Seminar on complex multiplication (Institute of Advanced Study 1957/8), Springer 1966
  • Representations des groupes localement compacts , Springer, Lecture notes in mathematics 276, 1972
  • Introductions aux groupes arithmétiques , Paris: Hermann 1969
  • Editor with Nolan Wallach and co-author: Continuous cohomology, discrete subgroups and representations of reductive groups , Princeton 1980, 2nd edition AMS 2000 (Seminar in Princeton 1976/77)
  • Intersection cohomology , Basel: Birkhäuser 1984
  • Algebraic D-modules , Academic Press 1987
  • Essays on the history of Lie groups and algebraic groups , American Mathematical Society 2001
  • with Robert Friedman , John W. Morgan : Almost commuting elements in compact Lie groups , American Mathematical Society 2002
  • with Lizhen Ji: Compactifications of symmetric and locally symmetric spaces , Birkhäuser 2006

Some essays by Borel:

  • Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts , Annals of Mathematics, Volume 57, 1953, pp. 115–207 (dissertation)
  • with Friedrich Hirzebruch: Characteristic classes and homogeneous spaces , American Journal of Mathematics, Volume 80, 1958, pp. 458-538
  • with Jean-Pierre Serre: La théorème de Riemann-Roch , Bulletin de la Société Mathématique de France 1958
  • with W.Baily Compactification of arithmetic quotients of bounded symmetric domains , Annals of Mathematics, Volume 84, 1966, pp. 442-528
  • Groupes lineaires algebriques , Annals of Mathematics, Volume 64, 1956, pp. 20-82
  • On the development of Lie group theory , Mathematical Intelligencer, Volume 2, 1980, pp. 67-72
  • 25 years with Bourbaki 1949-1973 , Notices AMS, 1998, No. 3, pp. 373-380
  • Hermann Weyl and Lie groups , in K. Chandrasekharan Weyl centennary symposium , Springer 1985

literature

Web links

Footnotes and Sources

  1. Technical: Borel subgroup is the maximum Zariski-closed connected solvable algebraic subgroup.