Raoul Bott

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Raoul Bott
Raoul Bott 1986.jpeg
Raoul Bott in 1986
Born (1923-09-24)September 24, 1923
Budapest, Hungary
Died December 20, 2005(2005-12-20) (aged 82)
San Diego, California
Residence Czechoslovakia
United States
Canada
Nationality Hungarian American
Fields Mathematics
Institutions University of Michigan in Ann Arbor
Harvard University
Alma mater McGill University
Carnegie Mellon University
Doctoral advisor Richard Duffin
Doctoral students Edward B. Curtis
Harold Edwards
Robert MacPherson
Daniel Quillen
Stephen Smale
Notable awards Veblen Prize (1964)
Wolf Prize (2000)

Raoul Bott, ForMemRS (September 24, 1923 – December 20, 2005)[1] was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem.

Early life[edit]

Bott was born in Budapest, Hungary, the son of Margit Kovács and Rudolph Bott.[2] His father was of Austrian descent, and his mother was of Hungarian descent; Bott was raised a Catholic by his mother and stepfather.[3][4] Bott grew up in Czechoslovakia and spent his working life in the United States. His family emigrated to Canada in 1938, and subsequently he served in the Canadian Army in Europe during World War II.

Career[edit]

Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a Ph.D. in mathematics from Carnegie Mellon University in Pittsburgh in 1949. His thesis, titled Electrical Network Theory, was written under the direction of Richard Duffin. Afterward, he began teaching at the University of Michigan in Ann Arbor. Bott continued his study at the Institute for Advanced Study in Princeton.[5] He was a professor at Harvard University from 1959 to 1999. In 2005, Dr. Bott died in San Diego after a battle with cancer.

With Richard Duffin at Carnegie Mellon, Bott studied existence of electronic filters corresponding to given positive rational functions . In 1949 they proved[6] a fundamental theorem of filter synthesis. Duffin and Bott extended earlier work by Otto Brune that requisite functions of complex frequency s could be realized by a passive network of inductors and capacitors. The proof, relying on induction on the sum of the degrees of the polynomials in the numerator and denominator of the rational function, was published in Journal of Applied Physics, volume 20, page 816. In his 2000 interview[7] with Allyn Jackson of the American Mathematical Society, he explained that he sees "networks as discrete versions of harmonic theory", so his experience with network synthesis and electronic filter topology introduced him to algebraic topology.

Bott met Arnold S. Shapiro at the IAS and they worked together. He studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem (1956). In the course of this work, he introduced Morse–Bott functions, an important generalization of Morse functions.

This led to his role as collaborator over many years with Michael Atiyah, initially via the part played by periodicity in K-theory. Bott made important contributions towards the index theorem, especially in formulating related fixed-point theorems, in particular the so-called 'Woods Hole fixed-point theorem', a combination of the Riemann–Roch theorem and Lefschetz fixed-point theorem (it is named after Woods Hole, Massachusetts, the site of a conference at which collective discussion formulated it).[8][citation needed] The major Atiyah–Bott papers on what is now the Atiyah–Bott fixed-point theorem were written in the years up to 1968; they collaborated further in recovering in contemporary language Ivan Petrovsky on Petrovsky lacunas of hyperbolic partial differential equations, prompted by Lars Gårding. In the 1980s, Atiyah and Bott investigated gauge theory, using the Yang–Mills equations on a Riemann surface to obtain topological information about the moduli spaces of stable bundles on Riemann surfaces.

He is also well known in connection with the Borel–Bott–Weil theorem on representation theory of Lie groups via holomorphic sheaves and their cohomology groups; and for work on foliations.

He introduced Bott–Samelson varieties and the Bott residue formula for complex manifolds and the Bott cannibalistic class.

Awards[edit]

In 1964, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society. In 1983, he was awarded the Jeffery–Williams Prize by the Canadian Mathematical Society. In 1987, he was awarded the National Medal of Science.[9]

In 2000, he received the Wolf Prize. In 2005, he was elected an Overseas Fellow of the Royal Society of London.

Students[edit]

Bott had 20 Ph.D. students, including Stephen Smale, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert W. Brooks, Robin Forman and Kevin Corlette.

Publications[edit]

  • Bott, Raoul Raoul Bott: collected papers. Vol. 4. Mathematics related to physics. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1995. xx+485 pp. ISBN 0-8176-3648-X MR 1321890
  • Bott, Raoul Raoul Bott: collected papers. Vol. 3. Foliations. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1995. xxxii+610 pp. ISBN 0-8176-3647-1 MR 1321886
  • Bott, Raoul Raoul Bott: collected papers. Vol. 2. Differential operators. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1994. xxxiv+802 pp. ISBN 0-8176-3646-3MR 1290361
  • Bott, Raoul Raoul Bott: collected papers. Vol. 1. Topology and Lie groups. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1994. xii+584 pp. ISBN 0-8176-3613-7 MR 1280032
  • Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982. xiv+331 pp. ISBN 0-387-90613-4 MR 0658304[10]
  • Bott, Raoul Lectures on K(X). Mathematics Lecture Note Series W. A. Benjamin, Inc., New York-Amsterdam 1969 x+203 pp.MR 0258020

References[edit]

  1. ^ Atiyah, M. (2007). "Raoul Harry Bott. 24 September 1923 -- 20 December 2005: Elected ForMemRS 2005". Biographical Memoirs of Fellows of the Royal Society 53: 63. doi:10.1098/rsbm.2007.0006.  edit
  2. ^ [1]
  3. ^ http://www-history.mcs.st-and.ac.uk/Biographies/Bott.html
  4. ^ http://www.ams.org/notices/200605/fea-bott-2.pdf
  5. ^ Community of Scholars from Institute for Advanced Study
  6. ^ John H. Hubbard (2010) "The Bott-Duffin Synthesis of Electrical Circuits", pp 33 to 40 in A Celebration of the Mathematical Legacy of Raoul Bott, P. Robert Kotiuga editor, CRM Proceedings and Lecture Notes #50, American Mathematical Society
  7. ^ Notices of the AMS 48(4):374
  8. ^ [2]
  9. ^ National Science Foundation
  10. ^ Stasheff, James D. (1984). "Review: Differential forms in algebraic topology, by Raoul Bott and Loring W. Tu". Bull. Amer. Math. Soc. (N.S,) 10 (1): 117–121. 

External links[edit]