Michel Demazure (; born 2 March 1937) is a French mathematician. He made contributions in the fields of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, algebraic geometry, and

computer vision Computer vision is an Interdisciplinarity, interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate t ...

, and participated in the Nicolas Bourbaki collective. He has also been president of the

French Mathematical Society French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Fran ...

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 In mathematics, the ''Séminaire de Géométrie Algébrique du Bois Marie'' (''SGA'') was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ...

group scheme In mathematics, a group scheme is a type of object from algebraic geometry 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 ha ...

s at the

Institut des Hautes Études Scientifiques The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics. It is located in Bures-sur-Yvette, ju ...

near Paris from 1962 to 1964. Demazure obtained his doctorate from the

Université de Paris The University of Paris (french: link=no, Université de Paris), metonymically known as the Sorbonne (), was the leading university in Paris, France, active from 1150 to 1970, with the exception between 1793 and 1806 under the French Revolutio ...

in 1965 under Grothendieck's supervision, with a dissertation entitled ''Schémas en groupes reductifs''. He was

maître de conférence ''Maître'' (spelled ''Maitre'' according to post-1990 spelling rules) is a commonly used honorific for lawyers, judicial officers and notaries in France, Belgium, Switzerland and French-speaking parts of Canada. It is often written in its abbr ...

Strasbourg University The University of Strasbourg (french: Université de Strasbourg, Unistra) is a public research university located in Strasbourg, Alsace, France, with over 52,000 students and 3,300 researchers. The French university traces its history to the e ...

(1964–1966), and then university professor at Paris-Sud in

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 centre of Paris. A fortified location of the Chevreuse valley since the 8th century a ...

(1966–1976)Author biography
from ''Bifurcations and Catastrophes''. and the

École Polytechnique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine The Seine ( , ) is a river in northern Franc ...

Palaiseau Palaiseau () is a commune in the southern suburbs of Paris, France. It is located from the centre of Paris. Palaiseau is a sub-prefecture of the Essonne department and the seat of the Arrondissement of Palaiseau. Inhabitants of Palaiseau ar ...

(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 Lactalis is a French multinational dairy products corporation, owned by the Besnier family and based in Laval, Mayenne, France. The company's former name was Besnier SA. Lactalis is the largest dairy products group in the world, and is the sec ...

. 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 The Cité des Sciences et de l'Industrie ("City of Science and Industry", abbreviated la CSI) or simply CSI is the biggest science museum in Europe. Located in the Parc de la Villette in Paris, France, it is one of the three dozen French Cultur ...

in La Villette, two major science museums in France; in taking these positions, he changed places with

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". From 1993 to 1996 he was president of the Cité des Sciences et de l'Industrie in La Villette, ...

, 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 (; oc, Lengadòc-Rosselhon ; ca, Llenguadoc-Rosselló) is a former administrative region of France. On 1 January 2016, it joined with the region of Midi-Pyrénées to become Occitania. It comprised five departments, and b ...

Research contributions

In SGA3, Demazure introduced the definition of a

root datum In mathematical group theory, 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. They were introduced by Michel Demazure in SGA III, publishe ...

, a generalization of

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, especially the classification and representat ...

s for

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

s that is central to the notion of

Langlands dual In representation theory, a branch of mathematics, the Langlands dual ''L'G'' of a reductive algebraic group ''G'' (also called the ''L''-group of ''G'') is a group that controls the representation theory of ''G''. If ''G'' is defined over a ...

ity. A 1970 paper of Demazure on

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

s of the

Cremona group In algebraic geometry, the Cremona group, introduced by , is the group of birational automorphisms of the n-dimensional projective space over a field It is denoted by Cr(\mathbb^n(k)) or Bir(\mathbb^n(k)) or Cr_n(k). The Cremona group is naturall ...

has been later recognized as the beginning of the study of

toric varieties In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be ...

. The

Demazure character formula 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. The Demazure character formula, introduced by , gives the charac ...

and Demazure modules and

Demazure conjecture In mathematics, the Demazure conjecture is a conjecture about representations of algebraic groups over the integers made by . The conjecture implies that many of the results of his paper can be extended from complex algebraic groups to algebraic g ...

are named after Demazure, who wrote about them in 1974. Demazure modules are submodules of a finite-dimensional representation of a

semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...

, and the Demazure character formula is an extension of the

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 . There is a closely related formula for the char ...

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 Gershevich (Grigorievich) Kac (russian: link=no, Виктор Гершевич (Григорьевич) Кац; born 19 December 1943) is a Soviet and American mathematician at MIT, known for his work in representation theory. He co-disco ...

, 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 Iterative reconstruction refers to iterative algorithms used to reconstruct 2D and 3D images in certain imaging techniques. For example, in computed tomography an image must be reconstructed from projections of an object. Here, iterative reco ...

problems in

. 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 The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative foca ...

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, . , . *''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. .

References

{{DEFAULTSORT:Demazure, Michel 1937 births Living people École Normale Supérieure alumni Algebraic geometers 20th-century French mathematicians 21st-century French mathematicians University of Paris alumni Nicolas Bourbaki