Michel Demazure
Encyclopedia
Michel Demazure is a French mathematician. He made contributions in the fields of abstract algebra
and algebraic geometry
, was president of the French Mathematical Society
and directed two French science museums.
s at the Institut des Hautes Études Scientifiques
near Paris from 1962 to 1964. Demazure obtained his doctorate from the Université de Paris in 1965 under Grothendieck's supervision, with a dissertation entitled Schémas en groupes reductifs. He was maître de conférence
at Strasbourg University (1964–1966), and then university professor at Paris-Sud
in Orsay
(1966–1976) and the École Polytechnique
in Palaiseau
(1976–1999). From approximately 1965 to 1985, he was also one of the core members of the Bourbaki group, a group of French mathematicians writing under the collective pseudonym Nicolas Bourbaki
.
In 1988 Demazure was the president of the Société Mathématique de France
.
From 1991 to 1998, he was the director of the Palais de la Découverte
in Paris and, from 1998 to 2002, the chairman of the Cité des Sciences et de l'Industrie
in La Villette, two major science museums in France; in taking these positions, he changed places with Jean Audouze
, who was at La Villette from 1993 to 1996, and became director of the Palais de la Découverte on Demazure's departure. Demazure also chairs the regional advisory committee of research for Languedoc-Roussillon
.
, a generalization of root system
s for reductive group
s that is central to the notion of Langlands duality.
The Demazure character formula and Demazure module
s are both named after Demazure, who wrote about them in 1974. Demazure modules are submodules of a finite dimensional representation of a semisimple Lie algebra, and the Demazure character formula is an extension of the Weyl character formula
to these modules. Demazure's work in this area was marred by a dependence on a false lemma in an earlier paper (also by Demazure); the flaw was pointed out by Victor Kac
, and subsequent research clarified the conditions under which the formula remains valid.
Later in his career, Demazure's research emphasis shifted from pure mathematics to more computational problems, involving the application of algebraic geometry to image reconstruction problems in computer vision
. The Kruppa–Demazure theorem, stemming from this work, shows that if a scene consisting of five points is viewed from two cameras with unknown positions but known focal length
s then, in general, there will be exactly ten different scenes that could have generated the same two images. Austrian mathematician Erwin Kruppa had many years earlier narrowed the number of possible scenes to eleven, and Demazure provided the first complete solution to the problem.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
and 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...
, was president of the French Mathematical Society
Société Mathématique de France
The Société Mathématique de France is the main professional society of French mathematicians.The society was founded in 1872 by Émile Lemoine and is one of the oldest mathematical societies in existence...
and directed two French science museums.
Biography
In the 1960s, Demazure was a student of Alexandre Grothendieck, and, together with Grothendieck, he ran and edited the Séminaire de Géométrie Algébrique du Bois Marie on group schemeGroup scheme
In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...
s at the Institut des Hautes Études Scientifiques
Institut des Hautes Études Scientifiques
The Institut des Hautes Études Scientifiques is a French institute supporting advanced research in mathematics and theoretical physics...
near Paris from 1962 to 1964. Demazure obtained his doctorate from the Université de Paris in 1965 under Grothendieck's supervision, with a dissertation entitled Schémas en groupes reductifs. He was maître de conférence
Academic rank in France
The following summarizes basic academic ranks in the French higher education system:-State university system:* PU equivalent to Full Professor)...
at Strasbourg University (1964–1966), and then university professor at Paris-Sud
Paris-Sud 11 University
University of Paris-Sud or University of Paris-Sud or University of Paris XI is a French university distributed among several campuses in the southern suburb of Paris...
in Orsay
Orsay
Orsay is a commune in the Essonne department in Île-de-France in northern France. It is located in the southwestern suburbs of Paris, France, from the center of Paris.Inhabitants of Orsay are known as Orcéens.-History:...
(1966–1976) and the École Polytechnique
École Polytechnique
The École Polytechnique is a state-run institution of higher education and research in Palaiseau, Essonne, France, near Paris. Polytechnique is renowned for its four year undergraduate/graduate Master's program...
in Palaiseau
Palaiseau
Palaiseau is a commune in the southern suburbs of Paris, France. It is located from the center of Paris. Palaiseau is a sub-prefecture of the Essonne department and the seat of the Arrondissement of Palaiseau....
(1976–1999). From approximately 1965 to 1985, he was also one of the core members of the Bourbaki group, a group of French mathematicians writing under the collective pseudonym Nicolas 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...
.
In 1988 Demazure was the president of the Société Mathématique de France
Société Mathématique de France
The Société Mathématique de France is the main professional society of French mathematicians.The society was founded in 1872 by Émile Lemoine and is one of the oldest mathematical societies in existence...
.
From 1991 to 1998, he was the director of the Palais de la Découverte
Palais de la Découverte
The Palais de la Découverte is a science museum located in the Grand Palais, in the 8th arrondissement on Avenue Franklin D. Roosevelt, Paris, France. It is open daily except Monday; an admission fee is charged....
in Paris and, from 1998 to 2002, the chairman of the Cité des Sciences et de l'Industrie
Cité des Sciences et de l'Industrie
The Cité des Sciences et de l'Industrie is the biggest science museum in Europe. Located in Parc de la Villette in Paris, France, it is at the heart of the Cultural Center of Science, Technology and Industry , a center promoting science and science culture.About five million people visit the Cité...
in La Villette, two major science museums in France; in taking these positions, he changed places with Jean Audouze
Jean Audouze
Jean Audouze is a French astrophysicist. He is a research director at CNRS and teaches at the Paris Institute of Political Science "Sciences po"...
, who was at La Villette from 1993 to 1996, and became director of the Palais de la Découverte on Demazure's departure. Demazure also chairs the regional advisory committee of research for Languedoc-Roussillon
Languedoc-Roussillon
Languedoc-Roussillon is one of the 27 regions of France. It comprises five departments, and borders the other French regions of Provence-Alpes-Côte d'Azur, Rhône-Alpes, Auvergne, Midi-Pyrénées on the one side, and Spain, Andorra and the Mediterranean sea on the other side.-Geography:The region is...
.
Research contributions
In SGA3, Demazure introduced the definition of a root datumRoot datum
In mathematics, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism...
, a generalization of root system
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
s for reductive group
Reductive group
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...
s that is central to the notion of Langlands duality.
The Demazure character formula and Demazure module
Demazure module
In mathematics, a Demazure module, introduced by . is a submodule of a finite dimensional representation generated by an extremal weight space under the action of a Borel subalgebra...
s are both named after Demazure, who wrote about them in 1974. Demazure modules are submodules of a finite dimensional representation of a semisimple Lie algebra, and the Demazure character formula is an extension of the Weyl character formula
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
to these modules. Demazure's work in this area was marred by a dependence on a false lemma in an earlier paper (also by Demazure); the flaw was pointed out by Victor Kac
Victor Kac
Victor G. Kac is a Soviet and American mathematician at MIT, known for his work in representation theory. He discovered Kac–Moody algebras, and used the Weyl–Kac character formula for them to reprove the Macdonald identities...
, and subsequent research clarified the conditions under which the formula remains valid.
Later in his career, Demazure's research emphasis shifted from pure mathematics to more computational problems, involving the application of algebraic geometry to image reconstruction problems in computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
. The Kruppa–Demazure theorem, stemming from this work, shows that if a scene consisting of five points is viewed from two cameras with unknown positions but known focal length
Focal length
The focal length of an optical system is a measure of how strongly the system converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus...
s then, in general, there will be exactly ten different scenes that could have generated the same two images. Austrian mathematician Erwin Kruppa had many years earlier narrowed the number of possible scenes to eleven, and Demazure provided the first complete solution to the problem.
Books
- Schémas en groupes. I: Propriétés générales des schémas en groupes (SGA3, vol. I, with Grothendieck). Lecture Notes in Mathematics 151, Berlin: Springer-Verlag, 1970. .
- Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux (SGA3, vol. II, with Grothendieck). Lecture Notes in Mathematics 152, Berlin: Springer-Verlag, 1970. .
- Schémas en groupes. III: Structure des schémas en groupes réductifs (SGA3, vol. III, with Grothendieck). Lecture Notes in Mathematics 153, Berlin: Springer-Verlag, 1970. .
- Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs (with Pierre Gabriel). Masson, Amsterdam: North Holland, 1970. . Partially translated into English by J. Bell as Introduction to Algebraic Geometry and Algebraic Groups, Volume 39 of North-Holland Mathematics Studies, Elsevier, 1980, .
- Lectures on p-divisible groups. Lecture Notes in Mathematics 302, Berlin: Springer-Verlag, 1972, 1986, ISBN 3-540-06092-8. , .
- Bifurcations and catastrophes: Geometry of solutions to nonlinear problems. Universitext, Berlin: Springer-Verlag, 2000. Translated from the French (1989) by David Chillingworth. .
- Cours d'Algèbre: Primalité. Divisibilité. Codes. Paris: Cassini, 1997, 2008. .