Alexander Grothendieck

Alexander Grothendieck

Alexander Grothendieck

(1928 - ) German-born, French mathematician, one of the pioneers of topos theory, and the
‘new

algebraic geometry

and

number theory

’. In 1966, he was awarded the

Fields Medal

for fundamental contributions to mathematics (an

algebraic

proof of one of the

Riemann-Roch theorems

, previously conjectured), but boycotted the ceremony held in Moscow (USSR), (as further explained in the text). Two decades later, he declined the

Crafoord Prize

that was awarded to him and his student Pierre Deligne, because he didn’t want the money and because the award was in recognition of work he had done much earlier in his career. Some concepts named after him include the

Grothendieck group

, the

Grothendieck topology

, the

Grothendieck category

(

http://

planetmath

.org/GrothendieckCategory

) and the Grothendieck universe.

In 1949, Grothendieck worked on functional analysis with Jean Dieudonné at the University of Nancy in France; he was one of the ‘Nicolas Bourbaki’ group of mathematicians that included at various times: André Weil, Henri Cartan, Charles Ehresmann and J. Dieudonné; Alexandre Grothendieck’s doctoral thesis, supervised by his advisor Laurent Schwartz, was entitled “Produits tensoriels topologiques et espaces nucléaires”.

From 1953 to 1955, Grothendieck was visiting at the University of São Paulo, supported by the Centre National de la Recherche Scientifique. He returned to France in 1956, to resume at the Centre National de la Récherche Scientifique.

In 1960, he visited at the University of Kansas in the

United States

working on topology and geometry, supported by the Centre National de la Récherche Scientifique beginning with 1956.

From 1959 to 1970 he acted as the Chair and

effective

leader of the newly formed Institut des Hautes Études Scientifiques (IHES); the IHES years have been referred to as his ‘Golden Age’, when an entire new school of abstract mathematics flourished under Grothendieck’s leadership; thus, Grothendieck’s Séminaire de Géométrie Algèbrique

[

3

,

4

]

established IHES as the World’s Center of algebraic geometry during the 1960s, with Grothendieck as its driving force. He travelled widely across Europe, including Eastern Europe (such as the invited visit he made in the Summer of 1968 when he delivered a lecture at the School of Mathematics in Bucharest at the invitation c/o Acad. Prof. Dr. Miron Nicolescu of the Romanian Academy (

supported after 1866 by Prince Charles von Hohenzollern-Sigmaringen–later in 1881–King Carol I of Romania

), and across the World. He is, and was, a very strong pacifist with very high ideals and goals, of real honesty and also extreme modesty; Alex campaigned against the military built-up of the 1960s, which built-up almost ended up in total annihilation of the planet during the Cuban missile crisis, and this is the reason for which he declined to go to Moscow to collect his Fields Medal in 1966 as a protest against the very aggressive, rapid, and indeed extremely dangerous, build-up of nuclear missiles by the Soviets.

Alexander Grothendieck’s work during the ‘Golden Age’ period established unifying themes in: algebraic geometry, number theory,

topology

and functional/complex analysis. Grothendieck introduced his own ‘theory of schemes’ in the 1960’s which allowed two of André Weil’s number theory

conjectures

to be solved by Alex, but in a much more general context
than originally envisaged by André Weil. One of his former French students, then proved the third

Weil conjecture

by a mathematical ‘bypass’ and his former student collected the Crafoord prize instead of Alex who publicly declined the prize in a letter that was only partially published in the French magazine “Le Monde” (The World). His stated
reasons were many, but in essence, he wanted to draw the attention of the entire world to the fact that mathematical
work was used in peacetime for war-related purposes which he very strongly disapproved of, and also that the mathematical profession and community that he knew had serious internal problems related to how mathematicians’ work and results were both evaluated and rewarded (mostly in France of the late 1970’s and 80’s!).

He worked on the theory of topoi/toposes that are relevant not only to mathematical logic and

category theory

, but also to algebraic geometry, number theory,

computer

software/programming and institutional ontology classification and bioinformatics. He provided an algebraic proof of one of the Riemann-Roch theorems, algebraic definition of the

fundamental group

of a curve, the

definition

of the fundamental

functor

for a categorical Galois theory, the re-definition of

Abelian categories

, (as for example in the case of

A

⁢

b

⁢

5

categories

that carry his name-the Grothendieck and

local Grothendieck categories

), he outlined the

http://planetmath.org/?op=getobj&from=lec&id=78

‘Dessins d’ Enfants’

combinatorial topology

theory and much, much more. His Séminaires de Géometrie algèbriques alone are several

thousands

of pages in (typewritten) printed length, or close to 500 Mb in electronic format.

In the early 1970s he was a Visiting Professor at the Collège de France, then at Orsay and Montpellier.

He retired in 1988 as an Emeritus Professor from the (French) University of Montpellier, and currently lives in France.

References

1

Alexander Grothendieck. 1957, Sur quelque point d-algèbre homologique. ,

Tohoku Math. J.

,

9:

119-121.

2

Alexander Grothendieck and J. Dieudoné.: 1960, Eléments de geometrie algèbrique.,

Publ. Inst. des Hautes Etudes de Science

,

4

.

3

Alexander Grothendieck et al.,1971. Séminaire de Géométrie Algèbrique du Bois-Marie, Vol. 1–7, Berlin: Springer-Verlag.

4

Alexander Grothendieck. 1962. Séminaire de Géométrie Algèbrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Cohérents et Théorḿes de Lefschetz Locaux et Globaux. , pp.287. (with an additional contributed exposé by Mme. Michele Raynaud).

http://modular.fas.harvard.edu/sga/sga/2/index.html

Typewritten manuscript available in French;

http://planetmath.org/?op=getobj&from=books&id=78

see also a brief summary in English

5

Alexander Grothendieck, 1984. “Esquisse d’un Programme”, (1984 manuscript), finally published in “Geometric Galois Actions”, L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp.5-48; English transl., ibid., pp. 243-283. MR 99c:14034 .

Title

Alexander Grothendieck

Canonical

name

AlexanderGrothendieck

Date of creation

2013-03-22 16:37:37

Last modified on

2013-03-22 16:37:37

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PrimeFan (13766)

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Numerical id

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Author

PrimeFan (13766)

Entry type

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msc 01A60

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