Armand Borel

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

Life and work

Borel studied at the ETH Zurich in the topologist Heinz Hopf and Eduard Stiefel, where he was assistant 1947-1949. In 1950 he was a professor in Geneva. From 1949 he was in Paris with Henri Cartan and Jean Leray with a CNRS scholarship, where he applied the spectral sequences of Jean Leray to 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 structure group. The cohomology these spaces provide the characteristic classes, such as in the case of the unitary groups Chern classes.

In France, he was also a member of the Bourbaki - circle, for whose book on Lie groups it is characterized mainly responsible. This book differs significantly in its abundance of "concrete" details from the others mostly very abstract Bourbaki volumes.

After he had been in Princeton 1952-1954 ( with a stopover in Chicago in 1954 with André Weil and from 1955 to 1957 as a professor at ETH ), where he worked among others with Friedrich Hirzebruch, he was from 1957-1993 professor at the Institute for Advanced Study in Princeton. In addition, he was 1983-1986 Professor at the ETH and also had numerous visiting professorships, for example, in India the Tata Institute of Fundamental Research in Bombay (1961, 1983, 1990) and in Hong Kong 1999-2001.

In addition to his work in algebraic topology and the theory of Lie groups he worked in algebraic groups, where he worked among others with Jacques Tits, and with arithmetic groups ( inter alia cooperation with Harish - Chandra ). His work on algebraic groups mid-1950s changed the whole area and allowed Claude Chevalley semisimple groups over arbitrary algebraically closed fields to classify. With Friedrich Hirzebruch in the case of the unitary group and generally with André Weil, he showed that the character formulas from Hermann Weyl for the irreducible representations of connected compact Lie groups G from the set of Hirzebruch - Riemann -Roch result, applied to the ( algebraic ) group ratio G / T (T = maximum torus of G), which is the fiber in the fiber bundle of the corresponding classifying space of G and T. On the fibers of the Weyl group of the Lie algebra surgery ( Vertauschungsgruppe the roots ), which is the symmetric group in the case of the unitary group, with a corresponding decomposition of the fiber in flag manifolds. Named after Borel Borel subgroup H of an algebraic group is defined by the homogeneous space G / H as is projective and so "small" possible. Example: = space of upper triangular matrices, where H is a maximal solvable subgroup and the " parabolic groups " P between H and G is the flag manifolds ( flag manifolds ) form G = general linear group GL ( n ), H.

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

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

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

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

Occasionally equivariant homology is called the Borel homology.

The Baily - Borel compactification in the theory of algebraic geometry is named after him and Walter Baily. She makes reference to specific arithmetic groups symmetric quotient spaces compact ( complete, completed ) and viewed with modular forms.

After Borel various conjectures are named. The Borel conjecture in topology is named after him. It arose from a question he asked in 1953 and Serre states that closed manifolds whose higher homotopy groups vanish ( aspherical manifolds ) and their fundamental groups are isomorphic, topologically equivalent ( homeomorphic ) are. The conjecture is open. Another assumption concerns the calculation of the Borel complex arithmetic cohomology group is given by the assumption of specific automorphous functions. It was proved by Jens Franke.

Borel was very interested in music and organized inter alia concerts with Indian and jazz music.

In 1992 he was awarded the Balzan Prize. In 1991 he received the Leroy P. Steele Prize of 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 on the ICM in Vancouver ( Cohomology of arithmetic groups). In 1978 he received the Brouwer Medal.