Alexander Grothendieck’s biography with his major mathematical contributions

Alexander Grothendieck’s biography with his major mathematical contributions

0.1

Alexander Grothendieck’s Biography and His Major Mathematical Contributions

Born:

March 28th, 1928 in Berlin, Germany

0.1.1

The Beginnings

A concise quote from an

http://www-groups.dcs.st-andrews.ac.uk/ history/Biographies/Grothendieck.html

article by J J O’Connor and E F Robertson is:

0.1.2

Quotes

“

Alexander Grothendieck

’s father was Russian and he (Alex’s father) was murdered by the Nazis.”

… (His mother, Hanka Grothendieck, was German);
…“Grothendieck moved to France in 1941 and later entered Montpellier University. After graduating from Montpellier he spent the year 1948-49 at the

É

cole Normale Sup

é

rieure in Paris.”

…“people are accustomed to work with

fundamental groups

and

generators and relations

for these and stick to it, even in

contexts

when this is wholly inadequate, namely when you get a clear description by generators and relations only when working simultaneously with a whole bunch of

base-points

chosen with care - or equivalently working in the

algebraic

context of

groupoids

, rather than groups. Choosing paths for connecting the base points natural to the situation to one among them, and reducing the groupoid to a single group, will then hopelessly destroy the

structure

and inner

symmetries

of the situation, and result in a mess of generators and relations no one dares to write down, because everyone feels they won’t be of any use whatever, and just confuse the picture rather than clarify it. I have known such perplexity myself a long time ago, namely in Van Kampen type situations, whose only understandable formulation is in terms of (

amalgamated sums

of) groupoids.”

by Alexander Grothendieck.

0.1.3

The Functional Analysis Phase

•

1949 Alex Grothendieck worked on

functional analysis

with Jean Dieudonn

é

e at the
University of Nancy in France; he was only for a short time one of the
‘Nicolas Bourbaki’ group (

http://

planetmath

.org/NicolasBourbaki

) of mathematicians that included at various times: Andr

é

Weil, Henri Cartan, Charles Ehresmann and Dieudonn

é

e. A quote from

http://www.ams.org/notices/200808/tx080800930p.pdf

“Who Is Grothendieck ?”:
“

To begin with, (L) Schwartz gave Grothendieck a paper to read that he had just written with Dieudonn

é

e, which ended with a list of fourteen unsolved problems. After a few months, Grothendieck had solved all of them. Try to visualize the situation: on one side, Schwartz, who had just received a

Fields Medal

and was at the top
of his scientific career, and on the other side the unknown student from the provinces, who had
a rather inadequate and unorthodox education. Grothendieck was awarded a Ph.D. for his work
on

topological vector spaces

and stuck with that field for a while.”

•

Alexander Grothendieck’s doctoral thesis supervised by his advisor Laurent Schwartz, and co-advised
by Jean Dieudonn

é

e was entitled “Produits tensoriels topologiques et espaces nucl

é

aires”;

0.1.4

Academic Appointments

•

1953-1955 Visiting at the University of S

ã

o Paulo, supported by the Centre National de la R

é

cherche Scientifique;

•

1956 Returned to France at the Centre National de la R

é

cherche Scientifique;

•

1960: Visiting at the University of Kansas in the USA working on topology and geometry, supported by the Centre National de la Recherche Scientifique beginning with 1956.

•

1970-72 Visiting Professor at Coll

é

ge de France.

•

1972-73 Visiting Professor at Orsay.

•

1973 Professor at the University of Montpellier.

•

1984-88 On leave– to direct research at the Centre National de la Recherche Scientifique.

0.1.5

His Golden Age at IHES

1959-1970: Chair of the newly formed Institut des Hautes

É

tudes 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 extremely creative leadership; thus,
Grothendieck’s S

é

minaire de G

é

om

é

trie Alg

è

brique

[

8

,

6

]

established
IHES as the

World’s Center of

Algebraic Geometry

during 1960-1970, with Alex as its
driving force. He travelled widely across Europe, including the Soviet-occupied 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 of Acad. Prof. Dr. Miron Nicolescu of the Romanian Academy
(

supported from 1866 by Prince Charles von Hohenzollern-Sigmaringen–who became in 1881–King Carol I of Romania

), and across the World. Grothendieck is a very strong pacifist with very high ideals and goals, of real honesty and also extreme modesty; Alexander Grothendieck campaigned against the military built-up of the 1960s, which built-up almost ended up in total annihilation of our planet during the Cuban missile crisis.

Alexander Grothendieck’s work during the ‘Golden Age’ period established unifying themes in:
Algebraic Geometry, Number theory,

Topology

,

Category Theory

and

Functional

/Complex Analysis.
He introduced his own ‘theory of schemes’ in the 1960’s which allowed certain of A. Weil’s number theory

conjectures

to be solved. He worked on the theory of topoi/toposes that are relevant not only to mathematical logic and category theory, but also to

computer

software/programming and institutional

ontology

classification and bioinformatics.
He provided an algebraic proof of the

Riemann-Roch theorem

, 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

‘Dessins d’ Enfants’

combinatorial topology

theory and much, much more. His
“S

é

⁢

m

inaires de G

é

ometrie alg

è

briques” alone are several

thousands

of pages in (typewritten) printed length, or close to 500 Mb in electronic format. Later in the ’80’s in his

‘Esquisse d’un Programme’

he outlined
the ‘anabelian’

homology

theory, what is called today in different fields by different names:

Non-Abelian

Homology Theory (that has not yet been achieved as he planned to do), non-Abelian

Algebraic Topology

,

Noncommutative geometry

, Non-Abelian

Quantum Field theories

, or ultimately,
non-Abelian

Categorical Ontology

, fields that are still in need of future developments.

0.1.6

Honors and Awards

•

Speaker at the

International Congress of Mathematicians

in 1958;

•

Alexander Grothendieck received the Fields Medal in 1966, which he accepted;

•

Alexander Grothendieck was awarded, but declined, the

Crafoord Prize

in 1988;
the prize was instead accepted by one of his

http://www.genealogy.math.ndsu.nodak.edu/id.php?id=61289

French former students;

•

Emeritus Professor in 1988 on his 60th birthday.

0.1.7

Author’s direct, First-hand impressions of Alexander Grothendieck:

One was struck immediately upon meeting him by his generosity and the energy with which
Alex shared his ideas with colleagues and students, as well as the excitement that
he incited through his brilliantly clear lecturing style, thus inspiring others to share
in his excitement for all of Mathematics, not just some highly specialized subject,
as if they were ‘to set out to explore a completely new land, or white territory’

.

0.1.8

A Brief Summary of Alexander Grothendieck’s Best-Known Contributions to Mathematics:

•

Topological tensor products and

nuclear spaces

,

•

Sheaf cohomology

as

derived functors

, schemes,

K-theory

and Grothendieck-Riemann-Roch,

•

Étale

Cohomology

and the Cohomological

interpretation

of L-functions,

•

Crystalline cohomology,

•

Defining and constructing geometric objects

via

Representable Functors

,

•

Descent, fibred categories and stacks,

•

Grothendieck topologies

(sites) and topoi,

•

Derived categories

,

•

Formalisms

for local and global duality (the ’six

operations

’),

•

Motives and the ’yoga of weights’,

•

Tensor Categories and Motivic Galois Groups.

•

Proofs of two generalized Riemann-Roch-Grothendieck

theorems

conjectured by André Weil.

Note:

Alexander Grothendieck’s

http://www.genealogy.math.ndsu.nodak.edu/id.php?id=31245&fChrono=1

mathematical ‘genealogy’ is claimed to go back through many successive doctoral advisor generations from Laurent Schwartz
to Borel, Darboux,…, Simeon Poisson, Joseph Lagrange, Leonhard Euler, Bernoulli, Gottfried Leibniz (in 1666, with a 53,763-long

sequence

of ‘

descendants

’), Weigel and Christiaan Huygens, and the record finally stops at Ludolph van Ceulen at the Universiteit Leiden in 1607 AD!

A most valuable resource in Algebraic Geometry, “Ho- and Coho- mology”

:

http://books.google.com/books?hl=en&id=FBfygannPSUC&dq=Alexandre+Grothendieck&printsec=frontcover&source=web&ots=Rwmt1x2weX&sig=y330F6qaDddY6_zyIJJA0SU602U&sa=X&oi=book_result&resnum=9&ct=result

Grothendieck-Serre Correspondence–Bilingual Edn.

References

1

Winfried Scharlau:

http://www.ams.org/notices/200808/tx080800930p.pdf

“Who Is Alexander Grothendieck ?”

2

Alexander Grothendieck. 1971, Revetements Etales et Groupe Fondamental (SGA1),
chapter VI: Categories fibrees et descente,

Lecture Notes in Math.

224

, Springer–Verlag: Berlin.

3

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

Tohoku Math. J.

,

9:

119-121.

4

Alexander Grothendieck and J. Dieudoné.: 1960, Elements de geometrie algebrique.,

Publ. Inst. des Hautes Etudes de Science

,

4

.

5

Alexander Grothendieck et al.,1971. Seminaire de Geometrie Algebrique du Bois-Marie, Vol. 1–7, Berlin: Springer-Verlag.

6

Alexander Grothendieck. 1962. Seminaire de Geometrie Algebrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Coherents et Theoremes de Lefschetz Locaux et Globaux. , pp.287. (with an additional contributed expose 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
References Cited:

1.

J. P. Serre. 1964.

Cohomologie Galoisienne

, Springer-Verlag: Berlin.

2.

J. L. Verdier. 1965.

Algebre homologiques et Categories derivees

. North Holland Publ. Cie.

7

Alexander Grothendieck. 1957, Sur Quelques Points d’algebre homologique,

Tohoku Mathematics Journal

, 9, 119–221.

8

Alexander Grothendieck et al.

Seminaires en Geometrie Algebrique- 4

, Tome 1, Expose’e 1
(or the Appendix to Expose’e 1, by ‘N.

Bourbaki

’ for more detail and a large number of results.
AG4 is

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

freely available in French;
also available here is an extensive

http://planetmath.org/?op=getobj&from=books&id=158

Abstract in English.

9

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 .

10

Alexander Grothendieck, “La longue marche in a travers la theorie de Galois”

= “The Long March Towards/Across the Theory of Galois”

, 1981 manuscript, University of Montpellier preprint series 1996, edited by J. Malgoire.

11

Leila Schneps. 1994.

http://planetmath.org/?op=getobj&from=books&id=163

The Grothendieck Theory of Dessins d’Enfants.
(

London Mathematical Society

Lecture Note Series), Cambridge University Press, 376 pp.

12

David Harbater and Leila Schneps. 2000.

http://www.ams.org/tran/2000-352-07/S0002-9947-00-02347-3/home.html

Fundamental groups of moduli and the Grothendieck-Teichmuller group,

Trans. Amer. Math. Soc

. 352 (2000), 3117-3148.
MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.

Title

Alexander Grothendieck’s biography with his major mathematical contributions

Canonical

name

AlexanderGrothendiecksBiographyWithHisMajorMathematicalContributions

Date of creation

2013-03-22 18:21:29

Last modified on

2013-03-22 18:21:29

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bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

104

Author

bci1 (20947)

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Topic

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msc 01A65

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msc 01A61

Classification

msc 01A60

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