André Weil

André Weil (/ˈveɪ/; French: [ɑ̃dʁe vɛj]; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry.^[3] He was one of the most influential mathematicians of the twentieth century. His influence is due both to his original contributions to a remarkably broad spectrum of mathematical theories, and to the mark he left on mathematical practice and style, through some of his own works as well as through the Bourbaki group, of which he was one of the principal founders.

André Weil
Born	(1906-05-06)6 May 1906 Paris, France
Died	6 August 1998(1998-08-06) (aged 92) Princeton, New Jersey, U.S.
Education	University of Paris École Normale Supérieure
Known for	List Bergman–Weil formula Borel–Weil theorem Chern–Weil homomorphism Chern–Weil theory De Rham–Weil theorem Weil's explicit formula Hasse-Weil bound Hasse–Weil zeta function Hasse–Weil L-function Mordell–Weil group Mordell–Weil theorem Oka–Weil theorem Siegel–Weil formula Shafarevich–Weil theorem Taniyama–Shimura–Weil conjecture Weil algebra Weil–Brezin Map Weil–Châtelet group Weil cohomology Weil conjectures Weil conjecture on Tamagawa numbers Weil's criterion Weil–Deligne group scheme Weil distribution Weil divisor Weil group Weil height Weil number Weil pairing Weil–Petersson metric Weil reciprocity law Weil representation Weil restriction
Awards	Wolf Prize (1979) Leroy P. Steele Prize (1980) Barnard Medal for Meritorious Service to Science (1980) Kyoto Prize (1994) ForMemRS (1966)^[1]
Scientific career
Fields	Mathematics
Institutions	Lehigh University Universidade de São Paulo (1945–47) University of Chicago (1947–58) Institute for Advanced Study
Doctoral advisor	Jacques Hadamard Charles Émile Picard
Doctoral students	Pierre Cartier Harley Flanders William A. Howard Teruhisa Matsusaka Peter Swinnerton-Dyer^[2] Alexandre Augusto Martins Rodrigues

Life

André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the German Empire after the Franco-Prussian War in 1870–71. Simone Weil, who would later become a famous philosopher, was Weil's younger sister and only sibling. He studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at Aligarh Muslim University in India. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he had taught himself Sanskrit in 1920.^[4][5] After teaching for one year at Aix-Marseille University, he taught for six years at University of Strasbourg. He married Éveline de Possel (née Éveline Gillet) in 1937.^[6]

Weil was in Finland when World War II broke out; he had been traveling in Scandinavia since April 1939. His wife Éveline returned to France without him. Weil was arrested in Finland at the outbreak of the Winter War on suspicion of spying; however, accounts of his life having been in danger were shown to be exaggerated.^[7] Weil returned to France via Sweden and the United Kingdom, and was detained at Le Havre in January 1940. He was charged with failure to report for duty, and was imprisoned in Le Havre and then Rouen. It was in the military prison in Bonne-Nouvelle, a district of Rouen, from February to May, that Weil completed the work that made his reputation. He was tried on 3 May 1940. Sentenced to five years, he requested to be attached to a military unit instead, and was given the chance to join a regiment in Cherbourg. After the fall of France in June 1940, he met up with his family in Marseille, where he arrived by sea. He then went to Clermont-Ferrand, where he managed to join his wife Éveline, who had been living in German-occupied France.

In January 1941, Weil and his family sailed from Marseille to New York. He spent the remainder of the war in the United States, where he was supported by the Rockefeller Foundation and the Guggenheim Foundation. For two years, he taught undergraduate mathematics at Lehigh University, where he was unappreciated, overworked and poorly paid, although he did not have to worry about being drafted, unlike his American students. He quit the job at Lehigh and moved to Brazil, where he taught at the Universidade de São Paulo from 1945 to 1947, working with Oscar Zariski. Weil and his wife had two daughters, Sylvie (born in 1942) and Nicolette (born in 1946).^[6]

He then returned to the United States and taught at the University of Chicago from 1947 to 1958, before moving to the Institute for Advanced Study, where he would spend the remainder of his career. He was a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts,^[8] in 1954 in Amsterdam,^[9] and in 1978 in Helsinki.^[10] Weil was elected Foreign Member of the Royal Society in 1966.^[1] In 1979, he shared the second Wolf Prize in Mathematics with Jean Leray.

Work

Weil made substantial contributions in a number of areas, the most important being his discovery of profound connections between algebraic geometry and number theory. This began in his doctoral work leading to the Mordell–Weil theorem (1928, and shortly applied in Siegel's theorem on integral points).^[11] Mordell's theorem had an ad hoc proof;^[12] Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which would not be categorized as such for another two decades. Both aspects of Weil's work have steadily developed into substantial theories.

Among his major accomplishments were the 1940s proof of the Riemann hypothesis for zeta-functions of curves over finite fields,^[13] and his subsequent laying of proper foundations for algebraic geometry to support that result (from 1942 to 1946, most intensively). The so-called Weil conjectures were hugely influential from around 1950; these statements were later proved by Bernard Dwork,^[14] Alexander Grothendieck,^[15][16][17] Michael Artin, and finally by Pierre Deligne, who completed the most difficult step in 1973.^{[18][19][20][21][22]}

Weil introduced the adele ring^[23] in the late 1930s, following Claude Chevalley's lead with the ideles, and gave a proof of the Riemann–Roch theorem with them (a version appeared in his Basic Number Theory in 1967).^[24] His 'matrix divisor' (vector bundle avant la lettre) Riemann–Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles. The Weil conjecture on Tamagawa numbers^[25] proved resistant for many years. Eventually the adelic approach became basic in automorphic representation theory. He picked up another credited Weil conjecture, around 1967, which later under pressure from Serge Lang (resp. of Serre) became known as the Taniyama–Shimura conjecture (resp. Taniyama–Weil conjecture) based on a roughly formulated question of Taniyama at the 1955 Nikkō conference. His attitude towards conjectures was that one should not dignify a guess as a conjecture lightly, and in the Taniyama case, the evidence was only there after extensive computational work carried out from the late 1960s.^[26]

Other significant results were on Pontryagin duality and differential geometry.^[27] He introduced the concept of a uniform space in general topology, as a by-product of his collaboration with Nicolas Bourbaki (of which he was a Founding Father). His work on sheaf theory hardly appears in his published papers, but correspondence with Henri Cartan in the late 1940s, and reprinted in his collected papers, proved most influential. He also chose the symbol ∅, derived from the letter Ø in the Norwegian alphabet (which he alone among the Bourbaki group was familiar with), to represent the empty set.^[28]

Weil also made a well-known contribution in Riemannian geometry in his very first paper in 1926, when he showed that the classical isoperimetric inequality holds on non-positively curved surfaces. This established the 2-dimensional case of what later became known as the Cartan–Hadamard conjecture.

He discovered that the so-called Weil representation, previously introduced in quantum mechanics by Irving Segal and David Shale, gave a contemporary framework for understanding the classical theory of quadratic forms.^[29] This was also a beginning of a substantial development by others, connecting representation theory and theta functions.

Weil was a member of both the National Academy of Sciences^[30] and the American Philosophical Society.^[31]

As expositor

Weil's ideas made an important contribution to the writings and seminars of Bourbaki, before and after World War II. He also wrote several books on the history of number theory.

Beliefs

Hindu thought had great influence on Weil.^[32] He was an agnostic,^[33] and he respected religions.^[34]

Legacy

Asteroid 289085 Andreweil, discovered by astronomers at the Saint-Sulpice Observatory in 2004, was named in his memory.^[35] The official naming citation was published by the Minor Planet Center on 14 February 2014 (M.P.C. 87143).^[36]

Books

Mathematical works:

• Arithmétique et géométrie sur les variétés algébriques (1935)^[37]
• Sur les espaces à structure uniforme et sur la topologie générale (1937)^[38]
• L'intégration dans les groupes topologiques et ses applications (1940)
• Weil, André (1946), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, vol. 29, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1029-3, MR 0023093^[39]
• Sur les courbes algébriques et les variétés qui s'en déduisent (1948)
• Variétés abéliennes et courbes algébriques (1948)^[40]
• Introduction à l'étude des variétés kählériennes (1958)
• Discontinuous subgroups of classical groups (1958) Chicago lecture notes
• Weil, André (1967), Basic number theory., Die Grundlehren der mathematischen Wissenschaften, vol. 144, Springer-Verlag New York, Inc., New York, ISBN 3-540-58655-5, MR 0234930^[41]
• Dirichlet Series and Automorphic Forms, Lezioni Fermiane (1971) Lecture Notes in Mathematics, vol. 189^[42]
• Essais historiques sur la théorie des nombres (1975)
• Elliptic Functions According to Eisenstein and Kronecker (1976)^[43]
• Number Theory for Beginners (1979) with Maxwell Rosenlicht^[44]
• Adeles and Algebraic Groups (1982)^[45]
• Number Theory: An Approach Through History From Hammurapi to Legendre (1984)^[46]

Collected papers:

• Œuvres Scientifiques, Collected Works, three volumes (1979)
• Weil, André (March 2009). Œuvres Scientifiques / Collected Papers. Springer Collected Works in Mathematics (in English, French, and German). Vol. 1 (1926–1951) (2nd printing ed.). Springer. ISBN 978-3-540-85888-1.^[47]
• Weil, André (March 2009). Œuvres Scientifiques / Collected Papers. Springer Collected Works in Mathematics (in English, French, and German). Vol. 2 (1951-1964) (2nd printing ed.). Springer. ISBN 978-3-540-87735-6.
• Weil, André (March 2009). Œuvres Scientifiques / Collected Papers. Springer Collected Works in Mathematics (in English, French, and German). Vol. 3 (1964-1978) (2nd printing ed.). Springer. ISBN 978-3-540-87737-0.

Autobiography:

• French: Souvenirs d'Apprentissage (1991) ISBN 3-7643-2500-3. Review in English by J. E. Cremona.
• English translation: The Apprenticeship of a Mathematician (1992), ISBN 0-8176-2650-6 Review by Veeravalli S. Varadarajan; Review by Saunders Mac Lane

Memoir by his daughter:

• At Home with André and Simone Weil by Sylvie Weil, translated by Benjamin Ivry; ISBN 978-0-8101-2704-3, Northwestern University Press, 2010.^[48]

See also

• List of things named after André Weil
• Trench

References

1. Serre, J.-P. (1999). "Andre Weil. 6 May 1906 – 6 August 1998: Elected For.Mem.R.S. 1966". Biographical Memoirs of Fellows of the Royal Society. 45: 519. doi:10.1098/rsbm.1999.0034.
2. André Weil at the Mathematics Genealogy Project
3. Horgan, J (1994). "Profile: Andre Weil – The Last Universal Mathematician". Scientific American. 270 (6): 33–34. Bibcode:1994SciAm.270f..33H. doi:10.1038/scientificamerican0694-33.
4. Amir D. Aczel,The Artist and the Mathematician, Basic Books, 2009 pp. 17ff., p. 25.
5. Borel, Armand
6. Ypsilantis, Olivier (31 March 2017). "En lisant " Chez les Weil. André et Simone "". Retrieved 26 April 2020.
7. Osmo Pekonen: L'affaire Weil à Helsinki en 1939, Gazette des mathématiciens 52 (avril 1992), pp. 13–20. With an afterword by André Weil.
8. Weil, André. "Number theory and algebraic geometry." 30 August 2017 at the Wayback Machine In Proc. Intern. Math. Congres., Cambridge, Mass., vol. 2, pp. 90–100. 1950.
9. Weil, A. "Abstract versus classical algebraic geometry" (PDF). In: Proceedings of International Congress of Mathematicians, 1954, Amsterdam. Vol. 3. pp. 550–558. Archived (PDF) from the original on 9 October 2022.
10. Weil, A. "History of mathematics: How and why" (PDF). In: Proceedings of International Congress of Mathematicians, (Helsinki, 1978). Vol. 1. pp. 227–236. Archived (PDF) from the original on 9 October 2022.
11. A. Weil, L'arithmétique sur les courbes algébriques, Acta Math 52, (1929) p. 281–315, reprinted in vol 1 of his collected papers ISBN 0-387-90330-5 .
12. L.J. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc Cam. Phil. Soc. 21, (1922) p. 179
13. Weil, André (1949), "Numbers of solutions of equations in finite fields", Bulletin of the American Mathematical Society, 55 (5): 497–508, doi:10.1090/S0002-9904-1949-09219-4, ISSN 0002-9904, MR 0029393 Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil ISBN 0-387-90330-5
14. Dwork, Bernard (1960), "On the rationality of the zeta function of an algebraic variety", American Journal of Mathematics, American Journal of Mathematics, Vol. 82, No. 3, 82 (3): 631–648, doi:10.2307/2372974, ISSN 0002-9327, JSTOR 2372974, MR 0140494
15. Grothendieck, Alexander (1960), "The cohomology theory of abstract algebraic varieties", Proc. Internat. Congress Math. (Edinburgh, 1958), Cambridge University Press, pp. 103–118, MR 0130879
16. Grothendieck, Alexander (1995) [1965], "Formule de Lefschetz et rationalité des fonctions L", Séminaire Bourbaki, vol. 9, Paris: Société Mathématique de France, pp. 41–55, MR 1608788
17. Grothendieck, Alexander (1972), Groupes de monodromie en géométrie algébrique, I: Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Lecture Notes in Mathematics, vol. 288, Springer-Verlag, doi:10.1007/BFb0068688, ISBN 978-3-540-05987-5, MR 0354656
18. Deligne, Pierre (1971), "Formes modulaires et représentations l-adiques", Séminaire Bourbaki vol. 1968/69 Exposés 347–363, Lecture Notes in Mathematics, vol. 179, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058801, ISBN 978-3-540-05356-9
19. Deligne, Pierre (1974), "La conjecture de Weil. I", Publications Mathématiques de l'IHÉS, 43 (43): 273–307, doi:10.1007/BF02684373, ISSN 1618-1913, MR 0340258, S2CID 123139343
20. Deligne, Pierre, ed. (1977), , Lecture Notes in Mathematics (in French), vol. 569, Berlin: Springer-Verlag, doi:10.1007/BFb0091516, ISBN 978-0-387-08066-6, archived from the original on 15 May 2009
21. Deligne, Pierre (1980), "La conjecture de Weil. II", Publications Mathématiques de l'IHÉS, 52 (52): 137–252, doi:10.1007/BF02684780, ISSN 1618-1913, MR 0601520, S2CID 189769469
22. Deligne, Pierre; Katz, Nicholas (1973), Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, vol. 340, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0060505, ISBN 978-3-540-06433-6, MR 0354657
23. A. Weil, Adeles and algebraic groups, Birkhauser, Boston, 1982
24. Weil, André (1967), Basic number theory., Die Grundlehren der mathematischen Wissenschaften, vol. 144, Springer-Verlag New York, Inc., New York, ISBN 3-540-58655-5, MR 0234930
25. Weil, André (1959), Exp. No. 186, Adèles et groupes algébriques, Séminaire Bourbaki, vol. 5, pp. 249–257
26. Lang, S. "Some History of the Shimura-Taniyama Conjecture." Not. Amer. Math. Soc. 42, 1301–1307, 1995
27. Borel, A. (1999). "André Weil and Algebraic Topology" (PDF). Notices of the AMS. 46 (4): 422–427. Archived (PDF) from the original on 9 October 2022.
28. Miller, Jeff (1 September 2010). "Earliest Uses of Symbols of Set Theory and Logic". Jeff Miller Web Pages. Retrieved 21 September 2011.
29. Weil, A. (1964). "Sur certains groupes d'opérateurs unitaires". Acta Math. (in French). 111: 143–211. doi:10.1007/BF02391012.
30. "Andre Weil". www.nasonline.org. Retrieved 20 December 2021.
31. "APS Member History". search.amphilsoc.org. Retrieved 20 December 2021.
32. Borel, Armand. [1] (see also)[2]
33. Paul Betz; Mark Christopher Carnes, American Council of Learned Societies (2002). American National Biography: Supplement, Volume 1. Oxford University Press. p. 676. ISBN 978-0-19-515063-6. Although as a lifelong agnostic he may have been somewhat bemused by Simone Weil's preoccupations with Christian mysticism, he remained a vigilant guardian of her memory,...
34. I. Grattan-Guinness (2004). I. Grattan-Guinness, Bhuri Singh Yadav (ed.). History of the Mathematical Sciences. Hindustan Book Agency. p. 63. ISBN 978-81-85931-45-6. Like in mathematics he would go directly to the teaching of the Masters. He read Vivekananda and was deeply impressed by Ramakrishna. He had affinity for Hinduism. Andre Weil was an agnostic but respected religions. He often teased me about reincarnation in which he did not believe. He told me he would like to be reincarnated as a cat. He would often impress me by readings in Buddhism.
35. "289085 Andreweil (2004 TC244)". Minor Planet Center. Retrieved 11 September 2019.
36. "MPC/MPO/MPS Archive". Minor Planet Center. Retrieved 11 September 2019.
37. Ore, Oystein (1936). "Book Review: Arithmétique et Géométrie sur les Variétés Algébriques". Bulletin of the American Mathematical Society. 42 (9): 618–619. doi:10.1090/S0002-9904-1936-06368-8.
38. Cairns, Stewart S. (1939). "Review: Sur les Espaces à Structure Uniforme et sur la Topologie Générale, by A. Weil" (PDF). Bull. Amer. Math. Soc. 45 (1): 59–60. doi:10.1090/s0002-9904-1939-06919-X. Archived (PDF) from the original on 9 October 2022.
39. Zariski, Oscar (1948). "Review: Foundations of Algebraic Geometry, by A. Weil" (PDF). Bull. Amer. Math. Soc. 54 (7): 671–675. doi:10.1090/s0002-9904-1948-09040-1. Archived (PDF) from the original on 9 October 2022.
40. Chern, Shiing-shen (1950). "Review: Variétés abéliennes et courbes algébriques, by A. Weil". Bull. Amer. Math. Soc. 56 (2): 202–204. doi:10.1090/s0002-9904-1950-09391-4.
41. Weil, André (1974). Basic Number Theory. doi:10.1007/978-3-642-61945-8. ISBN 978-3-540-58655-5.
42. Weil, André (1971), Dirichlet Series and Automorphic Forms: Lezioni Fermiane, Lecture Notes in Mathematics, vol. 189, doi:10.1007/bfb0061201, ISBN 978-3-540-05382-8, ISSN 0075-8434
43. Weil, André (1976). Elliptic Functions according to Eisenstein and Kronecker. doi:10.1007/978-3-642-66209-6. ISBN 978-3-540-65036-2.
44. Weil, André (1979). Number Theory for Beginners. New York, NY: Springer New York. doi:10.1007/978-1-4612-9957-8. ISBN 978-0-387-90381-1.
45. Humphreys, James E. (1983). "Review of Adeles and Algebraic Groups by A. Weil". Linear & Multilinear Algebra. 14 (1): 111–112. doi:10.1080/03081088308817546.
46. Ribenboim, Paulo (1985). "Review of Number Theory: An Approach Through History From Hammurapi to Legendre, by André Weil" (PDF). Bull. Amer. Math. Soc. (N.S.). 13 (2): 173–182. doi:10.1090/s0273-0979-1985-15411-4. Archived (PDF) from the original on 9 October 2022.
47. Berg, Michael (1 January 2015). "Review of Œuvres Scientifiques - Collected Papers, Volume 1 (1926–1951)". MAA Reviews, Mathematical Association of America.
48. Audin, Michèle (2011). "Review: At Home with André and Simone Weil, by Sylvie Weil" (PDF). Notices of the AMS. 58 (5): 697–698. Archived (PDF) from the original on 9 October 2022.

External links

 

Wikiquote has quotations related to André Weil.

• André Weil, by A. Borel, Bull. AMS 46 (2009), 661–666.
• André Weil: memorial articles in the Notices of the AMS by Armand Borel, Pierre Cartier, Komaravolu Chandrasekharan, Shiing-Shen Chern, and Shokichi Iyanaga
• Image of Weil
• A 1940 Letter of André Weil on Analogy in Mathematics
• Ford Burkhart (10 August 1998). "Andre Weil, Who Reshaped Mathematics, Is Dead at 92". The New York Times. Retrieved 10 January 2008.
• Paul Hoffman (3 January 1999). "The lives they lived: Andre Weil; Numbers Man". The New York Times. Retrieved 23 January 2008.
• Artless innocents and ivory-tower sophisticates: Some personalities on the Indian mathematical scene – M. S. Raghunathan
• Varadaraja, V.S. (April 1999). "Book Review: The Apprenticeship of a Mathematician—Autobiography of André Weil" (PDF). Notices of the AMS. 46 (4): 448–456.
• La vie et l'oeuvre d'André Weil, by J-P. Serre, L'Ens. Math. 45 (1999),5–16.
• Correspondance entre Henri Cartan et André Weil (1928–1991), par Michèle Audin, Doc. Math. 6, Soc. Math. France, 2011.