Jean-Louis Verdier (; 2 February 1935 – 25 August 1989) was a
French
French may refer to:
* Something of, from, or related to France
** French language, which originated in France
** French people, a nation and ethnic group
** French cuisine, cooking traditions and practices
Arts and media
* The French (band), ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
who worked, under the guidance of his doctoral advisor
Alexander Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
, on
derived categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
and
Verdier duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of
Alexander Groth ...
. He was a close collaborator of Grothendieck, notably contributing to
SGA 4 his theory of
hypercovers and anticipating the later development of
étale homotopy by
Michael Artin
Michael Artin (; born 28 June 1934) is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology Mathematics Department, known for his contributions to algebraic geometry. and
Barry Mazur
Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in ...
, following a suggestion he attributed to
Pierre Cartier.
Saul Lubkin's related theory of
rigid hypercovers was later taken up by
Eric Friedlander
Eric Mark Friedlander (born January 7, 1944) is an American mathematician who is a professor of mathematics at University of Southern California. He is best known for his works in algebraic topology, algebraic geometry, algebraic K-theory and r ...
in his definition of the
étale topological type In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts:
* Étale morphism
** Formally étale morphism
* Étale cohomology
* Étale topology
* Étale fundamental group
* Étale group scheme
* Ét ...
.
Verdier was a student at the elite
École Normale Supérieure
École or Ecole may refer to:
* an elementary school in the French educational stages normally followed by Secondary education in France, secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing i ...
in Paris, and later became director of studies there, as well as a Professor at the
University of Paris VII
Paris Diderot University, also known as Paris 7 (), was a French university located in Paris, France. It was one of the inheritors of the historic University of Paris, which was split into 13 universities in 1970. Paris Diderot merged with Paris ...
. For many years he directed a joint seminar at the École Normale Supérieure with
Adrien Douady
Adrien Douady (; 25 September 1935 – 2 November 2006) was a French mathematician born in La Tronche, Isère. He was the son of Daniel Douady and Guilhen Douady.
Douady was a student of Henri Cartan at the École normale supérieure, and initi ...
. Verdier was a member of
Bourbaki Bourbaki(s) may refer to :
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* Charles-Denis Bourbaki (1816–1897), French general, son of Constantin Denis Bourbaki
* Colonel Constantin Denis Bourbaki (1787–1827), officer in the Greek War of Independence and serving in the ...
. In 1984 he was the president of the
Société Mathématique de France
Groupe Lactalis S.A. (doing business as 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 S.A.
Lactalis is the largest dairy pr ...
.
In 1976 Verdier developed a useful regularity condition on
stratified sets that the Chinese-Australian mathematician
Tzee-Char Kuo had previously shown implied the
Whitney conditions In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965.
A stratification of a topological space is a finite filtration by closed subsets ...
for
subanalytic set In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points (for example in Euclidean space) defined in a way broader than for semianalytic sets (roughly speaking, those satisfying conditions requiri ...
s (such as real or complex analytic varieties). Verdier called the condition (w) for
Whitney, as at the time he thought (w) might be equivalent to Whitney's condition (b). Real algebraic examples for which the Whitney conditions (b) hold but Verdier's condition (w) fails, were constructed by
David Trotman
David John Angelo Trotman (27 September 1951 - 26 April 2025) was a mathematician, with dual British and French nationality. He is a grandson of the poet and author Oliver W. F. Lodge and a great-grandson of the physicist Oliver Lodge. He work ...
who has obtained many geometric properties of (w)-regular stratifications. Work of Bernard Teissier, aided by Jean-Pierre Henry and Michel Merle at the
École Polytechnique
(, ; also known as Polytechnique or l'X ) is a ''grande école'' located in Palaiseau, France. It specializes in science and engineering and is a founding member of the Polytechnic Institute of Paris.
The school was founded in 1794 by mat ...
, led to the 1982 result that Verdier's condition (w) is equivalent to the Whitney conditions for complex analytic stratifications.
Verdier later worked on the theory of
integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
s.
[Olivier Babelon, Pierre Cartier, ]Yvette Kosmann-Schwarzbach
Yvette Kosmann-Schwarzbach (born 30 April 1941) is a French mathematician and professor.
Education and career
Kosmann-Schwarzbach obtained her doctoral degree in 1970 at the University of Paris under supervision of André Lichnerowicz on a dis ...
: ''Integrable systems. The Verdier memorial colloquium.'' Birkhäuser, Progress in Mathematics, 1993.
Bibliography
* Verdier's 1967 thesis, published belatedly in:
*:
:Part of it also appears in
SGA 4½ as the last chapter, "Catégories dérivées (état 0)".
*
''Integrable Systems, The Verdier Memorial Conference''(Actes du Colloque International de Luminy, 1991), Progress in Mathematics 115, edited by O. Babelon, P. Cartier,
Y. Kosmann-Schwarzbach, Birkhäuser, 1993.
See also
*
Artin–Verdier duality
References
External links
*
{{DEFAULTSORT:Verdier, Jean-Louis
1935 births
1989 deaths
École Normale Supérieure alumni
20th-century French mathematicians
University of Paris alumni
Academic staff of the University of Paris
Nicolas Bourbaki