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 ...

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, Croatia a ...

, 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 primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...

. 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 tot ...

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 ...

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 vast majority of the world's countries—including all of the great powers—forming two opposin ...

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 Universit ...

Montreal Montreal ( ; officially Montréal, ) is the List of the largest municipalities in Canada by population, second-most populous city in Canada and List of towns in Quebec, most populous city in the Provinces and territories of Canada, Canadian ...

, 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 Entertainment * '' PhD: Phantasy Degree'', a Korean comic series * ''Piled Higher and Deeper'', a web comic * Ph.D. (band), a 1980s British group ** Ph.D. (Ph.D. albu ...

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 ...

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 Richard James Duffin (1909 – October 29, 1996) was an American physicist, known for his contributions to electrical transmission theory and to the development of geometric programming and other areas within operations research. Education and ...

. 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 ...

Ann Arbor Anne, alternatively spelled Ann, is a form of the Latin female given name Anna (name), Anna. This in turn is a representation of the Hebrew Hannah (given name), Hannah, which means 'favour' or 'grace'. Related names include Annie (given name), ...

. 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 scholar ...

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 higher le ...

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 ...

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 List of United States cities by population, eigh ...

. With

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 Ca ...

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 compl ...

''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 Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...

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, meetings, ...

, he explained that he sees "networks as discrete versions of harmonic theory", so his experience with

network synthesis Network synthesis is a design technique for linear electrical circuits. Synthesis starts from a prescribed impedance function of frequency or frequency response and then determines the possible networks that will produce the required response. ...

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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. Bott met

Arnold S. Shapiro Arnold Samuel Shapiro (1921, Boston, Massachusetts – 1962, Newton, Massachusetts) was an American mathematician known for his eversion of the sphere and Shapiro's lemma. He also was the author of an article on Clifford algebras and periodicit ...

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 topolog ...

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 additio ...

s, using methods from

Morse theory 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 differentiabl ...

, leading to the

(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 differentiabl ...

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 the ...

, initially via the part played by periodicity in

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...

. 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 at ...

, 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 sys ...

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 t ...

Petrovsky lacuna In mathematics, a Petrovsky lacuna, named for the Russian mathematician I. G. Petrovsky, is a region where the fundamental solution of a linear hyperbolic partial differential equation vanishes. They were studied by who found topological ...

s of

hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...

s, 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 Swe ...

. 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 groups) ...

, 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 Eu ...

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 spac ...

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 Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geom ...

he worked on

Nevanlinna theory In the mathematical field of complex analysis, Nevanlinna theory is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl called it "one of the few great mathematical events of (the twentieth) century ...

, studied

holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a com ...

s over

complex analytic manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

s 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 th ...

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 ob ...

. He introduced Bott–Samelson varieties and the

Bott residue formula In mathematics, the Bott residue formula, introduced by , describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold. Statement If ''v'' is a holomorphic vector field on a compact complex manifold ''M'', t ...

for complex manifolds and the

Bott cannibalistic class In mathematics, the Bott cannibalistic class, introduced by , is an element \theta_k(V) of the representation ring of a compact Lie group that describes the action of the Adams operation In mathematics, an Adams operation, denoted ψ''k'' for natur ...

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 is ...

by the

. 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 presen ...

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 natio ...

. 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, re ...

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 facult ...

, 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 gra ...

, Robert MacPherson,

Robert W. Brooks Robert Wolfe Brooks (Washington, D.C., September 16, 1952 – Montreal, September 5, 2002) was a mathematician known for his work in spectral geometry, Riemann surfaces, circle packings, and differential geometry. He received his Ph.D. from H ...

Robin Forman Robin may refer to: Animals * Australasian robins, red-breasted songbirds of the family Petroicidae * Many members of the subfamily Saxicolinae (Old World chats), including: **European robin (''Erithacus rubecula'') ** Bush-robin **Forest r ...

, Rama Kocherlakota,

Susan Tolman Susan Tolman is an American mathematician known for her work in symplectic geometry. She is a professor of mathematics at the University of Illinois at Urbana–Champaign, and Lynn M. Martin Professorial Scholar at Illinois. Tolman earned her P ...

, András Szenes, Kevin Corlette, and

Eric Weinstein Eric Ross Weinstein (born October 26, 1965) is an American podcast host and a managing director of Thiel Capital. Education Weinstein received his PhD in mathematical physics from Harvard University in 1992 under the supervision of Raoul Bo ...

. 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 ho ...

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 s ...

#82. Springer-Verlag, New York-Berlin. xiv+331 pp. * 1969: ''Lectures on K(X)''. Mathematics Lecture Note Series

W. A. Benjamin Benjamin Cummings is a Imprint (trade name), publishing imprint of Pearson Education that specializes in science. Benjamin Cummings publishes Medicine, medical textbooks, anatomy and physiology laboratory manuals, biology and microbiology textbook ...

, New York-Amsterdam x+203 pp.

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