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Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
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 comple ...
, 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 city, capital and List of cities and towns of Hungary, most populous city of Hungary. It is the List of cities in the European Union by population within city limits, tenth-largest city in the European Union by popul ...
,
Hungary Hungary is a landlocked country in Central Europe. Spanning much of the Pannonian Basin, 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 and ...
, 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 Czechoslovakia ( ; Czech language, Czech and , ''Česko-Slovensko'') was a landlocked country in Central Europe, created in 1918, when it declared its independence from Austria-Hungary. In 1938, after the Munich Agreement, the Sudetenland beca ...
and spent his working life in the
United States The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...
. His family emigrated to
Canada Canada is a country in North America. Its Provinces and territories of Canada, ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, making it the world's List of coun ...
in 1938, and subsequently he served in the
Canadian Army The Canadian Army () is the command (military formation), 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 re ...
in
Europe Europe is a continent located entirely in the Northern Hemisphere and mostly in the Eastern Hemisphere. It is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, the Mediterranean Sea to the south, and Asia to the east ...
during
World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...
.


Career

Bott later went to college at
McGill University McGill University (French: Université McGill) is an English-language public research university in Montreal, Quebec, Canada. Founded in 1821 by royal charter,Frost, Stanley Brice. ''McGill University, Vol. I. For the Advancement of Learning, ...
in
Montreal Montreal is the List of towns in Quebec, largest city in the Provinces and territories of Canada, province of Quebec, the List of the largest municipalities in Canada by population, second-largest in Canada, and the List of North American cit ...
, where he studied
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
. He then earned a PhD in mathematics from
Carnegie Mellon University Carnegie Mellon University (CMU) is a private research university in Pittsburgh, Pennsylvania, United States. The institution was established in 1900 by Andrew Carnegie as the Carnegie Technical Schools. In 1912, it became the Carnegie Institu ...
in
Pittsburgh Pittsburgh ( ) is a city in Allegheny County, Pennsylvania, United States, and its county seat. It is the List of municipalities in Pennsylvania#Municipalities, second-most populous city in Pennsylvania (after Philadelphia) and the List of Un ...
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 The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...
in
Ann Arbor Ann Arbor is a city in Washtenaw County, Michigan, United States, and its county seat. The 2020 United States census, 2020 census recorded its population to be 123,851, making it the List of municipalities in Michigan, fifth-most populous cit ...
. Bott continued his study at the
Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
in Princeton. He was a professor at
Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...
from 1959 to 1999. In 2005 Bott died of
cancer Cancer is a group of diseases involving Cell growth#Disorders, abnormal cell growth with the potential to Invasion (cancer), invade or Metastasis, spread to other parts of the body. These contrast with benign tumors, which do not spread. Po ...
in
San Diego San Diego ( , ) is a city on the Pacific coast of Southern California, adjacent to the Mexico–United States border. With a population of over 1.4 million, it is the List of United States cities by population, eighth-most populous city in t ...
. 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-element model, lumped electronic components, as opposed to distributed-element filters. That ...
s 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
inductor An inductor, also called a coil, choke, or reactor, is a Passivity (engineering), passive two-terminal electronic component, electrical component that stores energy in a magnetic field when an electric current flows through it. An inductor typic ...
s and
capacitor In electrical engineering, a capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the condenser, a term st ...
s. 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 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 and electronic filter topology 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 at the IAS and they worked together. He studied the
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
of
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
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 comple ...
(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 differenti ...
s. 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). 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 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 ...
s, prompted by Lars Gårding. 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 under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
, using the Yang–Mills equations 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 (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
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 topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ...
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 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 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 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, ...
. In 1983, he was awarded the Jeffery–Williams Prize by the Canadian Mathematical Society. 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 science, behavior ...
. 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, r ...
.


Students

Bott had 35 PhD students, including Stephen Smale, Lawrence Conlon, Daniel Quillen, Peter Landweber, 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 Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
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 (parti ...
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) () 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 size (with va ...
#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 *
Parallelizable manifold In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist Smooth function, smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a Basis of a vector space, ...
* Thom's and Bott's proofs of the Lefschetz hyperplane theorem


References


External links

* * . * (By Loring W. Tu, January 4, 2002). * (''
The New York Times ''The New York Times'' (''NYT'') is an American daily newspaper based in New York City. ''The New York Times'' covers domestic, national, and international news, and publishes opinion pieces, investigative reports, and reviews. As one of ...
'', 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 Department of Mathematics 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 Hungarian emigrants to Canada Canadian emigrants to the United States Hungarian expatriates in Czechoslovakia