Raoul Bott

Early life

Bott was born in Budapest, Hungary, the son of Margit Kovács and Rudolph Bott. His father was of Austrian descent, and his mother was of Hungarian Jewish descent; Bott was raised a Catholic by his mother and stepfather in Bratislava, Czechoslovakia, now the capital of Slovakia. 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

Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a PhD 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. He was a professor at Harvard University from 1959 to 1999. In 2005 Bott died of cancer in San Diego.

With Richard Duffin at Carnegie Mellon, Bott studied existence of electronic filters corresponding to given positive-real functions. In 1949 they proved 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 relied on induction on the sum of the degrees of the polynomials in the numerator and denominator of the rational function. In his 2000 interview 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 (1957). 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).{{cite journal

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. With Chern he worked on Nevanlinna theory, studied holomorphic vector bundles over complex analytic manifolds and introduced the Bott-Chern classes, useful in the theory of Arakelov geometry and also to algebraic number theory.

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

Awards

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.

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

Students

Bott had 35 PhD students, including Stephen Smale, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert W. Brooks, Susan Tolman, and Eric Weinstein. Smale and Quillen won Fields Medals in 1966 and 1978 respectively.

Publications

• 1995: Collected Papers. Vol. 4. Mathematics Related to Physics. Edited by Robert MacPherson. Contemporary Mathematicians. Birkhäuser Boston, xx+485 pp.
• 1995: Collected Papers. Vol. 3. Foliations. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxii+610 pp.
• 1994: Collected Papers. Vol. 2. Differential Operators. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxiv+802 pp.
• 1994: Collected Papers. Vol. 1. Topology and Lie Groups. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xii+584 pp.
• 1982: (with Loring W. Tu) Differential Forms in Algebraic Topology. Graduate Texts in Mathematics #82. Springer-Verlag, New York-Berlin. xiv+331 pp.
• 1969: Lectures on K(X). Mathematics Lecture Note Series W. A. Benjamin, New York-Amsterdam x+203 pp.

References

References

1. (2007). "Raoul Harry Bott. 24 September 1923 -- 20 December 2005: Elected ForMemRS 2005". [[Biographical Memoirs of Fellows of the Royal Society]].
2. (1 January 1995). "Notable Twentieth-century Scientists: A-E". Gale Research.
3. "Raoul Bott".
4. Tu, Loring W.. (May 2006). "The Life and Works of Raoul Bott". [[Notices of the American Mathematical Society]].
5. "Community of Scholars". [[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. (1949-08-01). "Impedance Synthesis without Use of Transformers". Journal of Applied Physics.
8. Jackson, Allyn, "Interview with Raoul Bott", [[Notices of the American Mathematical Society]] 48 (2001), no. 4, 374–382.
9. R. Bott. (1985). "On some recent interactions between mathematics and physics". [[Canadian Mathematical Bulletin]].
10. (7 January 2025). "Jeffery-Williams Prize – CMS-SMC".
11. "The President's National Medal of Science: Recipient Details - NSF - National Science Foundation".
12. "Raoul H. Bott".
13. Stasheff, James D.. "Review: ''Differential forms in algebraic topology'', by Raoul Bott and Loring W. Tu". [[Bulletin of the American Mathematical Society]].