Jean-Louis Verdier (; 2 February 1935 – 25 August 1989) was a French
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, structure, space, models, and change.
History
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 SGA may refer to:
* Old Irish language (ISO 639-3 code)
* ''Schwarz-Gelbe Allianz'' (Black-Yellow Alliance), an Austrian political party
* Second-generation antipsychotics
* Séminaire de Géométrie Algébrique du Bois Marie, an influential mat ...
Michael Artin
Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Barry Mazur, 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 in Santurce, Puerto Rico) is an American mathematician who is working in algebraic topology, algebraic geometry, algebraic K-theory and representation theory.
Friedlander graduated from Swarthmore ...
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 In algebraic geometry, the étale topology is a Grot ...
.
Verdier was a student at the elite
École Normale Supérieure
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 flowing in région Île-de-France
* École, Savoi ...
in Paris, and later became director of studies there, as well as a Professor at the
University of Paris VII
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, t ...
. 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.
Douady was a student of Henri Cartan at the École normale supérieure, and initially worked in homological algebra. His thesis concerned deformations of complex ...
. Verdier was a member of Bourbaki. In 1984 he 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 ...
.
In 1976 Verdier developed a useful regularity condition on
stratified sets
Stratification may refer to:
Mathematics
* Stratification (mathematics), any consistent assignment of numbers to predicate symbols
* Data stratification in statistics
Earth sciences
* Stable and unstable stratification
* Stratification, or s ...
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 sets (such as real or complex analytic varieties). Verdier called the condition (w) for
Whitney Whitney may refer to:
Film and television
* ''Whitney'' (2015 film), a Whitney Houston biopic starring Yaya DaCosta
* ''Whitney'' (2018 film), a documentary about Whitney Houston
* ''Whitney'' (TV series), an American sitcom that premiered i ...
, 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 (born 27 September 1951) is 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 Sir Oliver Lodge. He works in an are ...
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
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 flowing in région Île-de-France
* École, Savoi ...
, 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: ''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
In mathematics, Artin–Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by , that generalizes Tate duality.
It shows that, as far as etale (or flat) cohomolo ...