Raoul Bott | |
---|---|

Born | |

Died | December 20, 2005 82) | (aged

Nationality | Hungarian American |

Alma mater | McGill University Carnegie Mellon University |

Awards | Veblen Prize (1964) Jeffery–Williams Prize (1983) National Medal of Science (1987) Steele Prize (1990) Wolf Prize (2000) ForMemRS (2005) |

Scientific career | |

Fields | Mathematics |

Institutions | University of Michigan in Ann Arbor Harvard University |

Doctoral advisor | Richard Duffin |

Doctoral students |

**Raoul Bott** (September 24, 1923 – December 20, 2005)^{ [1] } was a Hungarian-American 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.

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 Jewish 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.

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 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^{ [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 (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).^{ [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. In 1983 he spoke to the Canadian Mathematical Society in a talk he called "A topologist marvels at Physics".^{ [9] }

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.

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.^{ [10] }

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

Bott had 35 Ph.D. students, including Stephen Smale, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert W. Brooks, Robin Forman, Rama Kocherlakota, András Szenes, Kevin Corlette,^{ [11] } and Eric Weinstein.^{ [12] }^{ [13] }^{ [14] } Smale and Quillen won Fields Medals in 1966 and 1978 respectively.

- 1995:
*Collected Papers. Vol. 4. Mathematics Related to Physics*. Edited by Robert MacPherson. Contemporary Mathematicians. Birkhäuser Boston, xx+485 pp. ISBN 0-8176-3648-X MR 1321890 - 1995:
*Collected Papers. Vol. 3. Foliations*. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxii+610 pp. ISBN 0-8176-3647-1 MR 1321886 - 1994:
*Collected Papers. Vol. 2. Differential Operators*. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxiv+802 pp. ISBN 0-8176-3646-3 MR 1290361 - 1994:
*Collected Papers. Vol. 1. Topology and Lie Groups*. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xii+584 pp. ISBN 0-8176-3613-7 MR 1280032 - 1982: (with Loring W. Tu)
*Differential Forms in Algebraic Topology*. Graduate Texts in Mathematics #82. Springer-Verlag, New York-Berlin. xiv+331 pp. ISBN 0-387-90613-4 doi : 10.1007/978-1-4757-3951-0 MR 0658304^{ [15] } - 1969:
*Lectures on K(X)*. Mathematics Lecture Note Series W. A. Benjamin, New York-Amsterdam x+203 pp.MR 0258020

**Sir Michael Francis Atiyah** was a British-Lebanese mathematician specialising in geometry.

**Georg Friedrich Bernhard Riemann** was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.

In differential geometry, the **Atiyah–Singer index theorem**, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the **analytical index** is equal to the **topological index**. It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.

**Oscar Zariski** was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century.

**Israel Moiseevich Gelfand**, also written **Israïl Moyseyovich Gel'fand**, or **Izrail M. Gelfand** was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the first Wolf Prize, he was a Foreign Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University. Gelfand is also a 1994 MacArthur Fellow.

**Armand Borel** was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups.

**Harold Calvin Marston Morse** was an American mathematician best known for his work on the *calculus of variations in the large*, a subject where he introduced the technique of differential topology now known as Morse theory. The Morse–Palais lemma, one of the key results in Morse theory, is named after him, as is the Thue–Morse sequence, an infinite binary sequence with many applications. In 1933 he was awarded the Bôcher Memorial Prize for his work in mathematical analysis.

**Friedrich Ernst Peter Hirzebruch** ForMemRS was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as "the most important mathematician in Germany of the postwar period."

**Norman Levinson** was an American mathematician. Some of his major contributions were in the study of Fourier transforms, complex analysis, non-linear differential equations, number theory, and signal processing. He worked closely with Norbert Wiener in his early career. He joined the faculty of the Massachusetts Institute of Technology in 1937. In 1954, he was awarded the Bôcher Memorial Prize of the American Mathematical Society and in 1971 the Chauvenet Prize of the Mathematical Association of America for his paper *A Motivated Account of an Elementary Proof of the Prime Number Theorem*. In 1974 he published a paper proving that more than a third of the zeros of the Riemann zeta function lie on the critical line, a result later improved to two fifths by Conrey.

**Vijay Kumar Patodi** was an Indian mathematician who made fundamental contributions to differential geometry and topology. He was the first mathematician to apply heat equation methods to the proof of the index theorem for elliptic operators. He was a professor at Tata Institute of Fundamental Research, Mumbai (Bombay).

In mathematics, the **Atiyah–Bott fixed-point theorem**, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds *M*, which uses an elliptic complex on *M*. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears in the original Lefschetz fixed-point theorem.

**Clifford Henry Taubes** is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother, Gary Taubes, is a science writer.

**Michael Jerome Hopkins** is an American mathematician known for work in algebraic topology.

In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed

**Lesley Millman Sibner** was an American mathematician and professor of mathematics at Polytechnic Institute of New York University. She earned her Bachelors at City College CUNY in Mathematics. She completed her doctorate at Courant Institute NYU in 1964 under the joint supervision of Lipman Bers and Cathleen Morawetz. Her thesis concerned partial differential equations of mixed-type.

**Harold Mortimer Edwards, Jr.** was an American mathematician working in number theory, algebra, and the history and philosophy of mathematics.

**Richard Sheldon Palais ** is a mathematician working in geometry who introduced the Principle of Symmetric Criticality, the Mostow–Palais theorem, the Lie–Palais theorem, the Morse–Palais lemma, and the Palais–Smale compactness condition.

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This is a **timeline of bordism**, a topological theory based on the concept of the boundary of a manifold. For context see timeline of manifolds. Jean Dieudonné wrote that cobordism returns to the attempt in 1895 to define homology theory using only (smooth) manifolds.

- ↑ Atiyah, Michael (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 . - ↑ McMurray, Emily J.; Kosek, Jane Kelly; Valade, Roger M. (1 January 1995).
*Notable Twentieth-century Scientists: A-E*. Gale Research. ISBN 9780810391826 . Retrieved 28 October 2016– via Internet Archive.Raoul Bott Margit Kovacs.

- ↑ "Bott biography" . Retrieved 28 October 2016.
- ↑ https://www.ams.org/notices/200605/fea-bott-2.pdf
- ↑ "Community of Scholars".
*ias.edu*. Institute for Advanced Study. Archived from the original on 2013-03-10. Retrieved 4 April 2018. - ↑ 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 - ↑ Jackson, Allyn, "Interview with Raoul Bott", Notices of the American Mathematical Society 48 (2001), no. 4, 374–382.
- ↑
^{[ }*dead link*] - ↑ R. Bott (1985). "On some recent interactions between mathematics and physics".
*Canadian Mathematical Bulletin*.**28**(2): 129–164. doi:10.4153/CMB-1985-016-3. - ↑ "The President's National Medal of Science: Recipient Details - NSF - National Science Foundation" . Retrieved 28 October 2016.
- ↑ "Raoul Bott - The Mathematics Genealogy Project" . Retrieved 28 October 2016.
- ↑ Eric Weinstein at the Mathematics Genealogy Project
- ↑ Tu, Loring W., ed. (2018). "Raoul Bott: Collected Papers, Volume 5".
*Notices of the American Mathematical Society*. Contemporary Mathematicians. Birkhäuser: 47. ISBN 9783319517810 . Retrieved 14 April 2020. - ↑ "PhD Dissertations Archival Listing".
*Harvard Mathematics Department*. Retrieved 2020-04-14. - ↑ Stasheff, James D. "Review:
*Differential forms in algebraic topology*, by Raoul Bott and Loring W. Tu".*Bulletin of the American Mathematical Society*. New Seriesyear=1984.**10**(1): 117–121. doi: 10.1090/S0273-0979-1984-15208-X .

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