# PlanetPhysics/Alexander Grothendieck

## Alexander Grothendieck

{\mathbf Born:} March 28th, 1928 in Berlin, Germany

A concise quote from an article by J J O'Connor and E F Robertson:

""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 \'Ecole Normale Sup\'erieure in Paris."

• 1949 Alex Grothendieck worked on functional analysis with Jean Dieudonn\'e 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: Andr\'e Weil, Henri Cartan, Charles Ehresmann and Jean Dieudonn\'e. A quote from "Who Is Grothendieck ?": {\em To begin with, (L) Schwartz gave Grothendieck a paper to read that he had just written with Dieudonn\'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\'e was entitled "Produits tensoriels topologiques et espaces nucl\'eaires" (in English: "Topological Tensor Products and Nuclear Spaces"; note that the concept of nuclear space has no connection to either atomic nuclei or nuclear weapons!).
• 1953-1955 Visiting at the University of S\~ao Paulo, supported by the Centre National de la Recherche Scientifique;
• 1956 Returned to France at the Centre National de la R\'echerche 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.
• 1959-1970: Chair of the newly formed Institut des Hautes \'Etudes 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\'eminaire de G\'eom\'etrie Alg\'ebrique [1] established IHES as the World's Center of \htmladdnormallink{algebraic {http://planetphysics.us/encyclopedia/CoIntersections.html} 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 (founded in 1866 by Prince Charles von Hohenzollern-Sigmaringen--who became in 1881--King Carol I of Romania ), and across the World. Alex is 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 our planet during the Cuban missile crisis. While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure, those who knew him say that the causes of the rupture ran deeper. Pierre Cartier, a visiteur de longue dur\'ee at the IH\'ES, wrote a piece about Grothendieck for a special volume published on the occasion of the IH\'ES's fortieth anniversary. In it Cartier notes that, as the son of an antimilitary anarchist and one who grew up among the disenfranchised, Grothendieck always had a deep compassion for the poor and the downtrodden . As Cartier puts it, Grothendieck came to find Bures--sur--Yvette "une cage dor\'ee" ("a golden cage"). While Grothendieck was at the IH\'ES, opposition to the Vietnam War was heating up, and Cartier suggests that this also reinforced Grothendieck's distaste at having become a mandarin of the scientific world. In addition, after several years at the IH\'ES Grothendieck seemed to cast about for new intellectual interests. By the late 1960s he had started to become interested in scientific areas outside of mathematics. David Ruelle, a physicist who joined the IH\'ES faculty in 1964, said that Grothendieck came to talk to him a few times about physics. Biology interested Grothendieck more than physics, and he organized some seminars on biological topics.

### Alex's Golden Age'

Alexander Grothendieck's work during the Golden Age' period established unifying themes in: Algebraic Geometry, Number Theory, Topology, category theory and Functional/Complex Analysis. Alex 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/ 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 $\displaystyle \A b5$ 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 S\'eminaires de G\'eometrie alg\ebriques 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.

• 1970-72 Visiting Professor at Coll\ege 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.

### 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 \htmladdnormallink{energy {http://planetphysics.us/encyclopedia/CosmologicalConstant.html} 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 some 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,
• \'Etale 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 Andr\'e 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.

## References

1. Cite error: Invalid <ref> tag; no text was provided for refs named Alexsem1, Alexsem2
2. ?[/itex]'> Winfried Scharlau: "Who Is Alexander Grothendieck ?"
3. Alexander Grothendieck. 1971, Rev\^{e}tements \'Etales et Groupe Fondamental (SGA1), chapter VI: Cat\'egories fibr\'ees et descente, Lecture Notes in Math. 224 , Springer--Verlag: Berlin.
4. Alexander Grothendieck. 1957, Sur quelque point d-alg\{e}bre homologique. , Tohoku Math. J. , 9: 119-121.
5. Alexander Grothendieck and J. Dieudon\'{e}.: 1960, El\'{e}ments de geometrie alg\{e}brique., Publ. Inst. des Hautes Etudes de Science , 4 .
6. Alexander Grothendieck et al.,1971. S\'eminaire de G\'eom\'etrie Alg\ebrique du Bois-Marie, Vol. 1--7, Berlin: Springer-Verlag.
7. Alexander Grothendieck. 1962. S\'eminaires en G\'eom\'etrie Alg\'ebrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Coh\erents et Th\'eor\emes de Lefschetz Locaux et Globaux. , pp.287. (with an additional contributed expos\'e 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. Alg\ebre homologiques et Cat\'egories deriv\'ees . North Holland Publ. Cie.
8. Alexander Grothendieck. 1957, Sur Quelques Points d'alg\ebre homologique, Tohoku Mathematics Journal , 9, 119--221.
9. Alexander Grothendieck et al. S\'eminaires en G\'eometrie Alg\ebrique- 4 , Tome 1, Expos\'e 1 (or the Appendix to Expos\'ee 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.
10. 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 {\mathbf 242}, Cambridge University Press, 1997, pp.5-48; English transl., ibid., pp. 243-283. MR 99c:14034 .
11. Alexander Grothendieck, "La longue marche in \`a travers la th\'eorie de Galois" = "The Long March Towards/Across the Theory of Galois" , 1981 manuscript, University of Montpellier preprint series 1996, edited by J. Malgoire.
12. Leila Schneps. 1994. The Grothendieck Theory of Dessins d'Enfants. (London Mathematical Society Lecture Note Series), Cambridge University Press, 376 pp.
13. David Harbater and Leila Schneps. 2000. Fundamental groups of moduli and the Grothendieck-Teichm\"uller group, Trans. Amer. Math. Soc . 352 (2000), 3117-3148. MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.