Raoul Bott
Encyclopedia
Raoul Bott, FRS was a Hungarian 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
.
, Hungary
, grew up in Slovakia
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
. He 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. He was a professor at Harvard University
from 1959 to 1999, and received the Wolf Prize in 2000. In 2005, he was elected an Overseas Fellow of the Royal Society of London. He died in San Diego after a battle with cancer.
Initially he worked on the theory of electrical circuits (Bott-Duffin theorem from 1949), then switched to pure mathematics.
He studied the homotopy theory of Lie group
s, using methods from Morse theory
, leading to the Bott periodicity theorem
(1956). 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 theorem
s, 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 results of Ivan Petrovsky
on hyperbolic partial differential equation
s, 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 space
s of stable bundles on Riemann surfaces.
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
s.
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
.
, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert Brooks, Robin Forman and Kevin Corlette.
. His Czech stepfather did not, so the principal language at home was German
. He had an English governess
from a young age, so he also spoke perfect English
(and retained a very faint English accent throughout his life). The language of his high school was Slovak
. Despite all this Bott claimed a distaste for learning languages.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
known for numerous basic contributions to geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
in its broad sense. He is best known for his Bott periodicity theorem
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 complex vector bundles, as well as the stable homotopy...
, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem
Borel–Bott–Weil theorem
In mathematics, the Borel–Weil-Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such...
.
Career
Bott was born in BudapestBudapest
Budapest is the capital of Hungary. As the largest city of Hungary, it is the country's principal political, cultural, commercial, industrial, and transportation centre. In 2011, Budapest had 1,733,685 inhabitants, down from its 1989 peak of 2,113,645 due to suburbanization. The Budapest Commuter...
, Hungary
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
, grew up in Slovakia
Slovakia
The Slovak Republic is a landlocked state in Central Europe. It has a population of over five million and an area of about . Slovakia is bordered by the Czech Republic and Austria to the west, Poland to the north, Ukraine to the east and Hungary to the south...
and spent his working life in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
. His family emigrated to Canada
Canada
Canada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...
in 1938, and subsequently he served in the Canadian Army
Canadian Forces
The Canadian Forces , officially the Canadian Armed Forces , are the unified armed forces of Canada, as constituted by the National Defence Act, which states: "The Canadian Forces are the armed forces of Her Majesty raised by Canada and consist of one Service called the Canadian Armed Forces."...
in Europe
Europe
Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...
during World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
. He later went to college at McGill University
McGill University
Mohammed Fathy is a public research university located in Montreal, Quebec, Canada. The university bears the name of James McGill, a prominent Montreal merchant from Glasgow, Scotland, whose bequest formed the beginning of the university...
in Montreal
Montreal
Montreal is a city in Canada. It is the largest city in the province of Quebec, the second-largest city in Canada and the seventh largest in North America...
, where he studied electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
. He then earned a Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
in mathematics from Carnegie Mellon University
Carnegie Mellon University
Carnegie Mellon University is a private research university in Pittsburgh, Pennsylvania, United States....
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
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
in Ann Arbor. He was a professor at Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
from 1959 to 1999, and received the Wolf Prize in 2000. In 2005, he was elected an Overseas Fellow of the Royal Society of London. He died in San Diego after a battle with cancer.
Initially he worked on the theory of electrical circuits (Bott-Duffin theorem from 1949), then switched to pure mathematics.
He studied the homotopy theory of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, using methods from Morse theory
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
, leading to the Bott periodicity theorem
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 complex vector bundles, as well as the stable homotopy...
(1956). 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
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
, initially via the part played by periodicity in K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
. Bott made important contributions towards the index theorem
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
, especially in formulating related fixed-point theorem
Fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point , under some conditions on F that can be stated in general terms...
s, in particular the so-called 'Woods Hole fixed-point theorem', a combination of the Riemann–Roch theorem
Riemann–Roch theorem
The Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles...
and Lefschetz fixed-point theorem
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 after Woods Hole, Massachusetts
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 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
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...
were written in the years up to 1968; they collaborated further in recovering in contemporary language results of Ivan Petrovsky
Ivan Petrovsky
Ivan Georgievich Petrovsky, also Petrovskii was a Soviet mathematician in the field of partial differential equations, and studied Petrovsky lacunas. He greatly contributed to solution of Hilbert's 19th and 16th problems...
on hyperbolic partial differential equation
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
s, prompted by Lars Gårding
Lars Gårding
Lars Gårding is a Swedish mathematician. He has made notable contributions to the study of partial differential operators. He is a professor emeritus of mathematics at Lund University in Sweden...
. In the 1980s, Atiyah and Bott investigated gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
, using the Yang–Mills equations on a Riemann surface to obtain topological information about the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s of stable bundles on Riemann surfaces.
He is also well-known in connection with the Borel–Bott–Weil theorem
Borel–Bott–Weil theorem
In mathematics, the Borel–Weil-Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such...
on representation theory of Lie groups via holomorphic sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
and their cohomology groups; and for work on foliation
Foliation
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
s.
Awards
In 1964, he was awarded the Oswald Veblen Prize in GeometryOswald Veblen Prize in Geometry
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...
by the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
. In 1983 he was awarded the Jeffery–Williams Prize
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 presented in 1968...
by the Canadian Mathematical Society
Canadian Mathematical Society
The Canadian Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and education in Canada.It was originally conceived in June 1945 as the Canadian Mathematical Congress...
. In 1987, he was awarded the National Medal of Science
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 sciences, biology, chemistry, engineering, mathematics and...
.
Students
Bott had 20 Ph.D. students, including Stephen SmaleStephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...
, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert Brooks, Robin Forman and Kevin Corlette.
Childhood
His mother and aunts spoke HungarianHungarian language
Hungarian is a Uralic language, part of the Ugric group. With some 14 million speakers, it is one of the most widely spoken non-Indo-European languages in Europe....
. His Czech stepfather did not, so the principal language at home was German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
. He had an English governess
Governess
A governess is a girl or woman employed to teach and train children in a private household. In contrast to a nanny or a babysitter, she concentrates on teaching children, not on meeting their physical needs...
from a young age, so he also spoke perfect English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
(and retained a very faint English accent throughout his life). The language of his high school was Slovak
Slovak language
Slovak , is an Indo-European language that belongs to the West Slavic languages .Slovak is the official language of Slovakia, where it is spoken by 5 million people...
. Despite all this Bott claimed a distaste for learning languages.
Publications
- Bott, Raoul Raoul Bott: collected papers. Vol. 4. Mathematics related to physics. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1995. xx+485 pp. ISBN 0-8176-3648-X
- Bott, Raoul Raoul Bott: collected papers. Vol. 3. Foliations. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1995. xxxii+610 pp. ISBN 0-8176-3647-1
- Bott, Raoul Raoul Bott: collected papers. Vol. 2. Differential operators. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1994. xxxiv+802 pp. ISBN 0-8176-3646-3
- Bott, Raoul Raoul Bott: collected papers. Vol. 1. Topology and Lie groups. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1994. xii+584 pp. ISBN 0-8176-3613-7
- Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982. xiv+331 pp. ISBN 0-387-90613-4
- Bott, Raoul Lectures on K(X). Mathematics Lecture Note Series W. A. Benjamin, Inc., New York-Amsterdam 1969 x+203 pp.
External links
- Commemorative website at Harvard Math Department
- "The Life and Works of Raoul Bott", by Loring Tu.
- "Raoul Bott, an Innovator in Mathematics, Dies at 82", The New York TimesThe New York TimesThe New York Times is an American daily newspaper founded and continuously published in New York City since 1851. The New York Times has won 106 Pulitzer Prizes, the most of any news organization...
, January 8, 2006.