Alexander Grothendieck's Biography and His Major Mathematical Contributions

The Beginnings

``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 École Normale Supérieure in Paris.''

The Functional Analysis Phase

• 1949 Alex Grothendieck worked on functional analysis with Jean Dieudonné at the University of Nancy in France; he was only for a short time one of the `Nicolas Bourbaki' group of mathematicians that included at various times: André Weil, Henri Cartan, Charles Ehresmann and Jean Dieudonné. A quote from ``Who Is Grothendieck ?'': ``To begin with, (L) Schwartz gave Grothendieck a paper to read that he had just written with Dieudonné, 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 Dieudonné was entitled ``Produits tensoriels topologiques et espaces nucléaires'';

Academic Appointments

• 1953-1955 Visiting at the University of São Paulo, supported by the Centre National de la Recherche Scientifique;
• 1956 Returned to France at the Centre National de la Récherche 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 Collége 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.

His Golden Age at IHES

1959-1970: Chair 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 extremely creative leadership; thus, Grothendieck's Séminaire de Géométrie Algèbrique [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 $\A b5$ 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 Séminaires de Géometrie algèbriques 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.

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 French former students;
• Emeritus Professor in 1988 on his 60th birthday.

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'.

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 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'': Grothendieck-Serre Correspondence-Bilingual Edn.

Bibliography

$?$_">1

Winfried Scharlau: ``Who Is Alexander Grothendieck ?''

2

Alexander Grothendieck. 1971, Revêtements Étales et Groupe Fondamental (SGA1), chapter VI: Catégories fibrées et descente, Lecture Notes in Math. 224, Springer-Verlag: Berlin.

3

Alexander Grothendieck. 1957, Sur quelque point d-algèbre homologique. , Tohoku Math. J., 9: 119-121.

4

Alexander Grothendieck and J. Dieudoné.: 1960, Eléments de geometrie algèbrique., Publ. Inst. des Hautes Etudes de Science, 4.

5

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

6

Alexander Grothendieck. 1962. Séminaire de Géométrie Algèbrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Cohérents et Théormes de Lefschetz Locaux et Globaux. , pp.287. (with an additional contributed exposé by Mme. Michele Raynaud). Typewritten manuscript available in French; see also a brief summary in English References Cited:

1. J. P. Serre. 1964. Cohomologie Galoisienne, Springer-Verlag: Berlin.
2. J. L. Verdier. 1965. Algèbre homologiques et Catégories derivées. North Holland Publ. Cie.

7

Alexander Grothendieck. 1957, Sur Quelques Points d'algébre homologique, Tohoku Mathematics Journal, 9, 119-221.

8

Alexander Grothendieck et al. Séminaires en Géometrie Algèbrique- 4, Tome 1, Exposé 1 (or the Appendix to Exposée 1, by `N. Bourbaki' for more detail and a large number of results. AG4 is freely available in French; also available here is an extensive 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 théorie 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. 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. Fundamental groups of moduli and the Grothendieck-Teichmüller group, Trans. Amer. Math. Soc. 352 (2000), 3117-3148. MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.