Jean-Louis Verdier
Encyclopedia
Jean-Louis Verdier was a French
mathematician
who worked, under the guidance of Alexander Grothendieck
, on derived categories
and Verdier duality
. He was a close collaborator of Alexander Grothendieck
, notably contributing to SGA 4 his theory of hypercovers and anticipating the later development of étale homotopy by Michael Artin
and 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 in his definition of the etale topological type.
Verdier was a student at the elite École Normale Supérieure
in Paris, and later became director of studies there, as well as a Professor at the University of Paris VII. For many years he directed a joint seminar at the Ecole Normale Supérieure
with Adrien Douady
.
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
for subanalytic set
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
hold but Verdier's condition (w) fails, were constructed by David Trotman
who has obtained many geometric properties of (w)-regular stratifications. Work of Bernard Teissier, aided by Jean-Pierre Henry and Michel Merle at the Ecole Polytechnique
, 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
s.
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...
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 worked, under the guidance 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...
, on derived categories
Derived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
and Verdier duality
Verdier duality
In mathematics, Verdier duality is a generalization of the Poincaré duality of manifolds to locally compact spaces with singularities. Verdier duality was introduced by , as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck...
. He was a close collaborator 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...
, notably contributing to SGA 4 his theory of hypercovers and anticipating the later development of étale homotopy by Michael Artin
Michael Artin
Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,...
and Barry Mazur
Barry Mazur
-Life:Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959,...
, following a suggestion he attributed to Pierre Cartier
Pierre Cartier (mathematician)
Pierre Cartier is a mathematician. An associate of the Bourbaki group and at one time a colleague of Alexander Grothendieck, his interests have ranged over algebraic geometry, representation theory, mathematical physics, and category theory....
. Saul Lubkin's related theory of rigid hypercovers was later taken up by Eric Friedlander in his definition of the etale topological type.
Verdier was a student at the elite École Normale Supérieure
École Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
in Paris, and later became director of studies there, as well as a Professor at the University of Paris VII. For many years he directed a joint seminar at the Ecole Normale Supérieure
École Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
with Adrien Douady
Adrien Douady
Adrien Douady was a French mathematician.He was a student of Henri Cartan at the Ecole Normale Supérieure, and initially worked in homological algebra. His thesis concerned deformations of complex analytic spaces...
.
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
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...
for subanalytic set
Subanalytic set
In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points defined in a way broader than for semianalytic sets...
s (such as real or complex analytic varieties). Verdier called the condition (w) for Whitney
Hassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
, as at the time he thought (w) might be equivalent to Whitney's condition (b). Real algebraic examples for which the Whitney conditions
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...
hold but Verdier's condition (w) fails, were constructed by David Trotman
David Trotman
David John Angelo Trotman is a mathematician, with dual British and French nationality. He was born on September 27, 1951 in Plymouth, Devon, England, a grandson of the poet and author Oliver W F Lodge and a great-grandson of the physicist Sir Oliver Lodge...
who has obtained many geometric properties of (w)-regular stratifications. Work of Bernard Teissier, aided by Jean-Pierre Henry and Michel Merle at the Ecole 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...
, led to the 1982 result that Verdier's condition (w) is equivalent to the Whitney conditions
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...
for complex analytic stratifications.
Verdier later worked on the theory of integrable system
Integrable system
In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...
s.