Raoul Bott (September 24, 1923 – December 20, 2005) was a
Hungarian-
American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the "United States" or "America"
** Americans, citizens and nationals of the United States of America
** American ancestry, pe ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
known for numerous basic contributions to
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
in its broad sense. He is best known for his
Bott periodicity theorem
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comp ...
, the
Morse–Bott functions which he used in this context, and the
Borel–Bott–Weil theorem.
Early life
Bott was born in
Budapest
Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union by population within city limits and the second-largest city on the Danube river; the city has an estimated population o ...
,
Hungary
Hungary ( hu, Magyarország ) is a landlocked country in Central Europe. Spanning of the Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to the northeast, Romania to the east and southeast, Serbia to the south, Cr ...
, 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. Bott grew up in
Czechoslovakia
, rue, Чеськословеньско, , yi, טשעכאסלאוואקיי,
, common_name = Czechoslovakia
, life_span = 1918–19391945–1992
, p1 = Austria-Hungary
, image_p1 ...
and spent his working life in the
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...
. His family emigrated to
Canada
Canada is a country in North America. Its ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, covering over , making it the world's second-largest country by to ...
in 1938, and subsequently he served in the
Canadian Army
The Canadian Army (french: Armée canadienne) is the command responsible for the operational readiness of the conventional ground forces of the Canadian Armed Forces. It maintains regular forces units at bases across Canada, and is also respo ...
in
Europe
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...
during
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...
.
Career
Bott later went to college at
McGill University
McGill University (french: link=no, Université McGill) is an English-language public research university located in Montreal, Quebec, Canada. Founded in 1821 by royal charter granted by King George IV,Frost, Stanley Brice. ''McGill Univer ...
in
Montreal
Montreal ( ; officially Montréal, ) is the second-most populous city in Canada and most populous city in the Canadian province of Quebec. Founded in 1642 as '' Ville-Marie'', or "City of Mary", it is named after Mount Royal, the triple- ...
, where he studied
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
. He then earned a
PhD PHD or PhD may refer to:
* Doctor of Philosophy (PhD), an academic qualification
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* Ph.D. (band), a 1980s British group
** Ph.D. (Ph.D. al ...
in mathematics from
Carnegie Mellon University
Carnegie Mellon University (CMU) is a private research university in Pittsburgh, Pennsylvania. One of its predecessors was established in 1900 by Andrew Carnegie as the Carnegie Technical Schools; it became the Carnegie Institute of Technology ...
in
Pittsburgh
Pittsburgh ( ) is a city in the Commonwealth (U.S. state), Commonwealth of Pennsylvania, United States, and the county seat of Allegheny County, Pennsylvania, Allegheny County. It is the most populous city in both Allegheny County and Wester ...
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
, mottoeng = "Arts, Knowledge, Truth"
, former_names = Catholepistemiad, or University of Michigania (1817–1821)
, budget = $10.3 billion (2021)
, endowment = $17 billion (2021)As o ...
in
Ann Arbor
Anne, alternatively spelled Ann, is a form of the Latin female given name Anna. This in turn is a representation of the Hebrew Hannah, which means 'favour' or 'grace'. Related names include Annie.
Anne is sometimes used as a male name in the ...
. Bott continued his study at the
Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
in Princeton. He was a professor at
Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of highe ...
from 1959 to 1999. In 2005 Bott died of
cancer
Cancer is a group of diseases involving abnormal cell growth with the potential to invade or spread to other parts of the body. These contrast with benign tumors, which do not spread. Possible signs and symptoms include a lump, abnormal b ...
in
San Diego
San Diego ( , ; ) is a city on the Pacific Ocean coast of Southern California located immediately adjacent to the Mexico–United States border. With a 2020 population of 1,386,932, it is the eighth most populous city in the United States ...
.
With
Richard Duffin at Carnegie Mellon, Bott studied existence of
electronic filter
Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components ...
s corresponding to given
positive-real function
Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis. They are complex functions, ''Z''(''s''), of a complex variable, ''s''. A rational function is ...
s. In 1949 they proved a fundamental theorem of
filter synthesis. Duffin and Bott extended earlier work by
Otto Brune
Otto Walter Heinrich Oscar Brune (10 January 1901 – 1982) undertook some key investigations into network synthesis at the Massachusetts Institute of Technology (MIT) where he graduated in 1929. His doctoral thesis was supervised by Wilhelm C ...
that requisite functions of
complex frequency
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
''s'' could be realized by a
passive network of
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s and
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s. 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
The ''Journal of Applied Physics'' is a peer-reviewed scientific journal with a focus on the physics of modern technology. The journal was originally established in 1931 under the name of ''Physics'', and was published by the American Physical So ...
, volume 20, page 816.
In his 2000 interview with Allyn Jackson of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
, he explained that he sees "networks as discrete versions of harmonic theory", so his experience with
network synthesis and
electronic filter topology
Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected.
Filter design characterises filter circuits primarily by their ...
introduced him to
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.
Bott met
Arnold S. Shapiro at the IAS and they worked together.
He studied the
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s, using methods from
Morse theory, leading to the
Bott periodicity theorem
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comp ...
(1957). In the course of this work, he introduced
Morse–Bott functions, an important generalization of
Morse function
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
s.
This led to his role as collaborator over many years with
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...
, initially via the part played by periodicity in
K-theory. Bott made important contributions towards the
index theorem
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
, especially in formulating related
fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors cla ...
s, in particular the so-called '
Woods Hole fixed-point theorem', a combination of the
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
and
Lefschetz fixed-point theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...
(it is named after
Woods Hole, Massachusetts
Woods Hole is a census-designated place in the town of Falmouth in Barnstable County, Massachusetts, United States. It lies at the extreme southwest corner of Cape Cod, near Martha's Vineyard and the Elizabeth Islands. The population was 781 ...
, the site of a conference at which collective discussion formulated it). The major Atiyah–Bott papers on what is now the
Atiyah–Bott fixed-point theorem
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 ...
were written in the years up to 1968; they collaborated further in recovering in contemporary language
Ivan Petrovsky
Ivan Georgievich Petrovsky (russian: Ива́н Гео́ргиевич Петро́вский) (18 January 1901 – 15 January 1973) (the family name is also transliterated as Petrovskii or Petrowsky) was a Soviet mathematician working mainly in ...
on
Petrovsky lacunas of
hyperbolic partial differential equations, prompted by
Lars Gårding
Lars Gårding (7 March 1919 – 7 July 2014) was a Swedish mathematician. He made notable contributions to the study of partial differential equations and partial differential operators. He was a professor of mathematics at Lund University in ...
. In the 1980s, Atiyah and Bott investigated
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...
, using the
Yang–Mills equations
In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the E ...
on a Riemann surface to obtain topological information about the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s 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".
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
foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
s. 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 In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Background
The main motivation behind Arakelov geometry is ...
and also to
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
.
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
__NOTOC__
The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen. The Veblen Prize is now worth US$5000, and ...
by the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
. In 1983, he was awarded the
Jeffery–Williams Prize The Jeffery–Williams Prize is a mathematics award presented annually by the Canadian Mathematical Society. The award is presented to individuals in recognition of outstanding contributions to mathematical research. The first award was present ...
by the
Canadian Mathematical Society
The Canadian Mathematical Society (CMS) (french: Société mathématique du Canada) is an association of professional mathematicians dedicated to the interests of mathematical research, outreach, scholarship and education in Canada. It serves the ...
. In 1987, he was awarded the
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social scienc ...
.
In 2000, he received the
Wolf Prize
The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of nati ...
. In 2005, he was elected an Overseas Fellow of the
Royal Society of London
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, r ...
.
Students
Bott had 35 PhD students, including
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty ...
,
Lawrence Conlon,
Daniel Quillen
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...
,
Peter Landweber
Peter Steven Landweber (born August 17, 1940, in Washington D. C.) is an American mathematician working in algebraic topology.
Landweber studied at the University of Iowa (B.SC. 1960) and the Harvard University (master's degree 1961), where he g ...
,
Robert MacPherson,
Robert W. Brooks,
Robin Forman,
Rama Kocherlakota,
Susan Tolman,
András Szenes,
Kevin Corlette, and
Eric Weinstein.
Smale and Quillen won
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
s in 1966 and 1978 respectively.
Publications
* 1995: ''Collected Papers. Vol. 4. Mathematics Related to Physics''. Edited by
Robert MacPherson. Contemporary Mathematicians.
Birkhäuser
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields:
* Springer continues to publish science (particu ...
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
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard ...
#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.
See also
*
Bott–Duffin inverse
In linear algebra, a constrained generalized inverse is obtained by solving a system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variab ...
*
Parallelizable manifold
*
Thom's and Bott's proofs of the Lefschetz hyperplane theorem
References
External links
*
Commemorative website at Harvard Math Department by Loring Tu.
''
The New York Times
''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid ...
'', January 8, 2006.
{{DEFAULTSORT:Bott, Raoul
1923 births
2005 deaths
20th-century American mathematicians
21st-century American mathematicians
American people of Hungarian-Jewish descent
Hungarian Jews
20th-century Hungarian mathematicians
Topologists
Geometers
Differential geometers
Algebraic geometers
Harvard University faculty
University of Michigan faculty
McGill University Faculty of Engineering alumni
Carnegie Mellon University alumni
Foreign Members of the Royal Society
National Medal of Science laureates
Wolf Prize in Mathematics laureates
Members of the French Academy of Sciences
Hungarian Roman Catholics
Deaths from cancer in California
Hungarian emigrants to Canada
Canadian emigrants to the United States
Hungarian expatriates in Czechoslovakia