Claude Chevalley
Encyclopedia
Claude Chevalley was a French
mathematician
who made important contributions to number theory
, algebraic geometry
, class field theory
, finite group theory, and the theory of algebraic groups. He was a founding member of the Bourbaki
group.
in 1929, where he studied under Émile Picard. He then spent time at the University of Hamburg
, studying under Emil Artin
, and at the University of Marburg, studying under Helmut Hasse
. In Germany, Chevalley discovered Japanese mathematics in the person of Shokichi Iyanaga
. Chevalley was awarded a doctorate in 1933 from the University of Paris
for a thesis on class field theory
.
When World War II
broke out, Chevalley was at Princeton University
. After reporting to the French Embassy, he stayed in the USA, first at Princeton then (after 1947) at Columbia University
. His American students included Leon Ehrenpreis
and Gerhard Hochschild
. During his time in the USA, Chevalley became an American citizen and wrote a substantial part of his lifetime output in English.
When Chevalley applied for a chair at the Sorbonne
, the difficulties he encountered were the subject of a polemical piece by his friend and fellow Bourbakiste André Weil
, titled "Science Française?" and published in the NRF. Chevalley was the "professeur B" of the piece, as confirmed in the endnote to the reprint in Weil's collected works, Oeuvres Scientifiques, tome II. Chevalley eventually did obtain a position in 1957 at the faculty of sciences of the University of Paris
, and after 1970 at the Université de Paris VII.
Chevalley had artistic and political interests, and was a minor member of the French non-conformists of the 1930s
. The following quote by the co-editor of Chevalley's collected works attests to these interests:
, removing a use of L-function
s and replacing it by an algebraic method. At that time use of group cohomology
was implicit, cloaked by the language of central simple algebra
s. In the introduction to André Weil
's Basic Number Theory, Weil attributed the book's adoption of that path to an unpublished manuscript by Chevalley.
Around 1950, Chevalley wrote a three-volume treatment of Lie group
s. A few years later, he published the work for which he is best remembered, his investigation into what are now called Chevalley groups. Chevalley groups make up 9 of the 18 families of finite simple groups.
Chevalley's accurate discussion of integrality conditions in the Lie algebra
s of semisimple groups enabled abstracting their theory from the real
and complex fields. As a consequence, analogues over finite field
s could be defined. This was an essential stage in the evolving classification of finite simple groups
. After Chevalley's work, the distinction between "classical groups" falling into the Dynkin diagram classification, and sporadic groups which did not, became sharp enough to be useful. What are called 'twisted' groups of the classical families could be fitted into the picture.
"Chevalley's theorem" (also called the Chevalley–Warning theorem) usually refers to his result on the solubility of equations over a finite field. Another theorem of his concerns the constructible set
s in algebraic geometry
, i.e. those in the Boolean algebra generated by the Zariski-open and Zariski-closed sets. It states that the image
of such a set by a morphism
of algebraic varieties
is of the same type. Logicians call this an elimination of quantifiers.
In the 1950s, Chevalley led some Paris seminars (working groups to the English speaking world) of major importance: the Séminaire Cartan–Chevalley of the academic year 1955/6, with Henri Cartan
, and the Séminaire Chevalley of 1956/7 and 1957/8. These dealt with topics on algebraic group
s and the foundations of algebraic geometry, as well as pure abstract algebra
. The Cartan–Chevalley seminar was the genesis of scheme theory, but its subsequent development in the hands of Alexander Grothendieck
was so rapid, thorough and inclusive that its historical tracks can appear well covered. Grothendieck's work subsumed the more specialised contribution of Serre
, Chevalley, Goro Shimura
, and others such as Erich Kähler
and Masayoshi Nagata
.
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician
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....
who made important contributions to number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
, finite group theory, and the theory of algebraic groups. He was a founding member of the Bourbaki
Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
group.
Life
Chevalley's father was a French diplomat who, jointly with his wife Marguerite, wrote The Concise Oxford French Dictionary. Chevalley graduated from the École Normale SupérieureÉcole Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
in 1929, where he studied under Émile Picard. He then spent time at the University of Hamburg
University of Hamburg
The University of Hamburg is a university in Hamburg, Germany. It was founded on 28 March 1919 by Wilhelm Stern and others. It grew out of the previous Allgemeines Vorlesungswesen and the Kolonialinstitut as well as the Akademisches Gymnasium. There are around 38,000 students as of the start of...
, studying under Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
, and at the University of Marburg, studying under Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
. In Germany, Chevalley discovered Japanese mathematics in the person of Shokichi Iyanaga
Shokichi Iyanaga
was a Japanese mathematician.-Early life:Iyanaga was born in Tokyo, Japan on April 2, 1906. He studied at the University of Tokyo from 1926 to 1929. He studied under Teiji Takagi. As an undergraduate, he published two papers in the Japanese Journal of Mathematics and the Proceedings of the Imperial...
. Chevalley was awarded a doctorate in 1933 from the University of Paris
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
for a thesis on class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
.
When 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...
broke out, Chevalley was at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. After reporting to the French Embassy, he stayed in the USA, first at Princeton then (after 1947) at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
. His American students included Leon Ehrenpreis
Leon Ehrenpreis
Leon Ehrenpreis was a mathematician at Temple University who proved the Malgrange–Ehrenpreis theorem, the fundamental theorem about differential operators with constant coefficients....
and Gerhard Hochschild
Gerhard Hochschild
Gerhard Paul Hochschild was an American mathematician who worked on Lie groups, algebraic groups, homological algebra and algebraic number theory....
. During his time in the USA, Chevalley became an American citizen and wrote a substantial part of his lifetime output in English.
When Chevalley applied for a chair at the Sorbonne
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
, the difficulties he encountered were the subject of a polemical piece by his friend and fellow Bourbakiste André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
, titled "Science Française?" and published in the NRF. Chevalley was the "professeur B" of the piece, as confirmed in the endnote to the reprint in Weil's collected works, Oeuvres Scientifiques, tome II. Chevalley eventually did obtain a position in 1957 at the faculty of sciences of the University of Paris
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
, and after 1970 at the Université de Paris VII.
Chevalley had artistic and political interests, and was a minor member of the French non-conformists of the 1930s
Non-conformists of the 1930s
The Non-Conformists of the 1930s refers to a nebula of groups and individuals during the inter-war period in France which was looking for new solutions to face the political, economical and social crisis. The name was coined in 1969 by the historian Jean-Louis Loubet del Bayle to describe a...
. The following quote by the co-editor of Chevalley's collected works attests to these interests:
"Chevalley was a member of various avant-garde groups, both in politics and in the arts... Mathematics was the most important part of his life, but he did not draw any boundary between his mathematics and the rest of his life."
Work
In his PhD thesis, Chevalley made an important contribution to the technical development of class field theoryClass field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
, removing a use of L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s and replacing it by an algebraic method. At that time use of group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
was implicit, cloaked by the language of central simple algebra
Central simple algebra
In ring theory and related areas of mathematics a central simple algebra over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K...
s. In the introduction to André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
's Basic Number Theory, Weil attributed the book's adoption of that path to an unpublished manuscript by Chevalley.
Around 1950, Chevalley wrote a three-volume treatment 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. A few years later, he published the work for which he is best remembered, his investigation into what are now called Chevalley groups. Chevalley groups make up 9 of the 18 families of finite simple groups.
Chevalley's accurate discussion of integrality conditions in the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s of semisimple groups enabled abstracting their theory from the real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
and complex fields. As a consequence, analogues over finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s could be defined. This was an essential stage in the evolving classification of finite simple groups
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
. After Chevalley's work, the distinction between "classical groups" falling into the Dynkin diagram classification, and sporadic groups which did not, became sharp enough to be useful. What are called 'twisted' groups of the classical families could be fitted into the picture.
"Chevalley's theorem" (also called the Chevalley–Warning theorem) usually refers to his result on the solubility of equations over a finite field. Another theorem of his concerns the constructible set
Constructible set (topology)
In topology, a constructible set in a noetherian topological space is a finite union of locally closed sets. Constructible sets form a Boolean algebra In fact, the constructible sets are precisely the...
s in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, i.e. those in the Boolean algebra generated by the Zariski-open and Zariski-closed sets. It states that the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
of such a set by a morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
of algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
is of the same type. Logicians call this an elimination of quantifiers.
In the 1950s, Chevalley led some Paris seminars (working groups to the English speaking world) of major importance: the Séminaire Cartan–Chevalley of the academic year 1955/6, with Henri Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
, and the Séminaire Chevalley of 1956/7 and 1957/8. These dealt with topics on algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s and the foundations of algebraic geometry, as well as pure abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. The Cartan–Chevalley seminar was the genesis of scheme theory, but its subsequent development in the hands of Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
was so rapid, thorough and inclusive that its historical tracks can appear well covered. Grothendieck's work subsumed the more specialised contribution of Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, Chevalley, Goro Shimura
Goro Shimura
is a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.Shimura was a colleague and a friend of Yutaka Taniyama...
, and others such as Erich Kähler
Erich Kähler
was a German mathematician with wide-ranging geometrical interests.Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig. He held professorial positions in Königsberg, Leipzig, Berlin and Hamburg...
and Masayoshi Nagata
Masayoshi Nagata
Masayoshi Nagata was a Japanese mathematician, known for his work in the field of commutative algebra....
.
Selected bibliography
- 1940, "La théorie du corps de classes," Annals of Mathematics 41: 394–418.
- 1946. Theory of Lie groups . Princeton University Press.
- 1951. "Théorie des groupes de Lie, tome II, Groupes algébriques", Hermann, Paris.
- 1951. Introduction to the theory of algebraic functions of one variable, A.M.S. Math. Surveys VI.
- 1954. The algebraic theory of spinors, Columbia Univ. Press; new edition, Springer-Verlag, 1997.
- 1953-1954. Class field theory, Nagoya Univ..
- 1955. "Théorie des groupes de Lie, tome III, Théorèmes généraux sur les algèbres de Lie", Hermann, Paris.
- 1955, "Sur certains groupes simples," Tôhoku Mathematical Journal 7: 14–66.
- 1955. The construction and study of certain important algebras, Publ. Math. Soc. Japan..
- 1956. Fundamental concepts of algebra, Acad. Press.
- 1956-1958. "Classification des groupes de Lie algébriques", Séminaire Chevalley, Secrétariat Math., 11 rue P. Curie, Paris; revised edition by P.Cartier, Springer-Verlag, 2005.
- 1958. Fondements de la géométrie algébrique, Secrétariat Math., 11 rue P. Curie, Paris.
See also
- idèle
- valuative criterion of properness
- Chevalley group
- Chevalley schemeChevalley schemeA Chevalley scheme in algebraic geometry was a precursor notion of scheme theory.Let X be a separated integral noetherian scheme, R its function field...
- Beck–Chevalley condition
- Non-Conformist Movement
- Jordan–Chevalley decomposition
External links
- Claude Chevalley in the Mathematics Genealogy ProjectMathematics Genealogy ProjectThe Mathematics Genealogy Project is a web-based database for the academic genealogy of mathematicians. As of September, 2010, it contained information on approximately 145,000 mathematical scientists who contribute to "research-level mathematics"...