name = André Weil
birth_date = birth date|1906|5|6
death_date = death date and age|1998|8|6|1906|5|6
Lehigh University Universidade de São Paulo University of Chicago Institute for Advanced Study
letter theory, algebraic geometry
Lion Prize in Mathematics
:André Weil "should not be confused with two other mathematicians with similar names:
Hermann Weyl(1885-1955), "who made substantial contributions to theoretical physicsand number theory. In 1944, he helped Weil obtain a Guggenheim fellowship.
Andrew Wiles(1953-), "whose proof of Fermat's Last Theoremrequired that he prove a conjecture partly due to Weil and who, like Weil, has done important work in elliptic curves;-----
André Weil (
May 6, 1906- August 6, 1998) (pronounced|ɑ̃dʁe vɛj [In English, "Weil" is pronEng|ˈveɪ "vay", while " Weyl" is IPA|/ˈvaɪl/ "vial" and "Wiles" is IPA|/ˈwaɪlz/ "why-ulz". However, in Russian the names Weil and Weyl are pronounced in the same way (as IPA| [vʲejlʲ] "vail"). ] ) was one of the greatest mathematicians of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theoryand algebraic geometry. He was a founding member and the "de facto" early leader of the influential Bourbaki group. The philosopher Simone Weilwas his sister.
Paristo Alsatian Jewish parents who fled the annexation of Alsace-Lorraineto Germany, Weil studied in Paris, Romeand Göttingenand received his doctorate in 1928. While in Germany, he befriended Carl Ludwig Siegel. He spent two academic years at Aligarh Muslim Universityfrom 1930. Sanskrit literaturewas a life-long interest. After one year in Marseille, he taught six years in Strasbourg. He married Eveline in 1937.
Weil was in
Finlandwhen World War IIbroke out; he had been traveling in Scandinavia since April 1939. Eveline returned to France without him. The following anecdote is taken from his autobiography: after having been arrested under suspicion of espionagein Finland, when the USSR attacked on 30 November 1939, he was saved from being shot only by the intervention of Rolf Nevanlinna. This is the version that Nevanlinna propagated after the war. However, such a story is a bit too good to be true. In 1992, the Finnish mathematician Osmo Pekonenwent to the archives to check the facts. Based on the documents, he established that Weil was not really going to be shot, even if he was under arrest, and that Nevanlinna probably didn't do - and didn't need to do - anything to save him. Pekonen published a paper [Osmo Pekonen: "L'affaire Weil à Helsinki en 1939", Gazette des mathématiciens 52 (avril 1992), pp. 13—20. With an afterword by André Weil.] on this with an afterword by André Weil himself. Nevanlinna's motivation for concocting such a story of himself as the rescuer of a famous Jewish mathematician probably was the fact that he had been a Nazi sympathizer during the war. The story also appears in Nevanlinna's autobiography, published in Finnish, but the dates don't match with real events at all. It is true, however, that Nevanlinna housed Weil in the summer of 1939 at his summer residence Korkee at Lohjain Finland - and offered Hitler's Mein Kampfas bedside reading. Weil signed ' Bourbaki' in Nevanlinna's guestbook.
Weil returned to France via Sweden and the United Kingdom, and was detained at
Le Havrein 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 he did the work that made his reputation. He was tried on May 3 1940. Sentenced to five years, he asked to be sent to a military unit instead, and joined a regiment in Cherbourg. After the fall of France, he met up with his family in Marseille, where he arrived by sea. He then went to Clermont-Ferrand, where he managed to join Eveline, who had been in German-occupied France.
In January 1941, Weil and his family sailed from
Marseilleto New York. He spent the war in the United States, where he was supported by the Rockefeller Foundationand Guggenheim Foundation. For two years, he unhappily taught undergraduate mathematics at Lehigh University. He taught at the Universidade de São Paulo, 1945-47, where he worked with Oscar Zariski. He taught at the University of Chicagofrom 1947 to 1958, before spending the remainder of his career at the Institute for Advanced Study. In 1979, he shared the second Wolf Prize in Mathematics.
He made substantial contributions in many areas, the most important being his discovery of profound connections between
algebraic geometryand 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). Mordell's theoremhad an "ad hoc" proof; Weil began the separation of the infinite descentargument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades. Both aspects have steadily developed into substantial theories.
Among his major accomplishments were the 1940 proof, while in prison, of the
Riemann hypothesisfor local zeta-functions, and his subsequent laying of proper foundations for algebraic geometry to support that result (from 1942 to 1946, most intensively). By modern standards his claim to have a proof had a very easy ride, but wartime conditions were one factor, and the fact that the German experts made little or no comment another. The so-called Weil conjectureswere hugely influential from around 1950; they were later proved by Bernard Dwork, Alexander Grothendieck, Michael Artin, and Pierre Deligne, who completed the most difficult step in 1973.
He had introduced the
adele ringin the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann-Roch theoremwith them (a version appeared in his "Basic Number Theory" in 1967). His 'matrix divisor' ( vector bundle"avant le jour") Riemann-Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles. The Weil conjecture on Tamagawa numbersproved resistant for many years. Eventually the adelic approach became basic in automorphic representationtheory. He picked up another credited "Weil conjecture", around 1970, which later under pressure from Serge Langbecame known as the Taniyama-Shimura conjecturebased on the presentation of the basic ideas at the 1955 Nikkō conference. His attitude towards conjectures struck many in the field as oblique; he wrote that one should not dignify a guess as a conjecture lightly, and in the Shimura-Taniyama case, the evidence was only there after extensive computational work carried out from the late 1960s.
Other significant results were on
Pontryagin dualityand differential geometry. He introduced the concept of uniform spacein general topology. His work on sheaf theoryhardly appears in his published papers, but correspondence with Henri Cartanin the late 1940s, and reprinted in his collected papers, proved most influential.
He discovered that the so-called
Weil representation, previously introduced in quantum mechanicsby Irving Segaland Shale, gave a contemporary framework for understanding the classical theory of quadratic forms. This was also a beginning of a substantial development by others, connecting representation theoryand theta-functions.
Weil's ideas made an important contribution to the writings and seminars of
Bourbaki, before and after World War II. His books had an important influence on research, and exceptional situation in mathematics. (In one famous case, the influence was possibly negative: Alexander Grothendieckis said to have complained of the 'aridity' of Weil's "Foundations of Algebraic Geometry".) The style of his books is clearly demarcated from that of his research papers.
He invented the notation "Ø" for the
*"Arithmétique et géométrie sur les variétés algébriques" (1935)
*"Sur les espaces à structure uniforme et sur la topologie générale" (1937)
*"L'intégration dans les groupes topologiques et ses applications" (1940)
*"Foundations of Algebraic Geometry" (1946)
*"Sur les courbes algébriques et les variétés qui s’en déduisent" (1948)
*"Variétés abéliennes et courbes algébriques" (1948)
*"Introduction à l'étude des variétés kählériennes" (1958)
*"Discontinuous subgroups of classical groups" (1958) Chicago lecture notes
*"Basic Number Theory" (1967)
*"Dirichlet Series and Automorphic Forms, Lezioni Fermiane" (1971) Lecture Notes in Mathematics, vol. 189,
*"Essais historiques sur la théorie des nombres" (1975)
*"Elliptic Functions According to Eisenstein and Kronecker" (1976)
*"Œuvres Scientifiques, Collected Works, three volumes" (1979)
*"Number Theory for Beginners" (1979) with Maxwell Rosenlicht
*"Adeles and Algebraic Groups" (1982)
*"Number Theory: An Approach Through History From Hammurapi to Legendre" (1984)
*French: "Souvenirs d’Apprentissage" (1991) ISBN 3764325003. [http://links.jstor.org/sici?sici=0025-5572(199307)2%3A77%3A479%3C261%3AAWSD%3E2.0.CO%3B2-B Review in English] by J. E. Cremona.
*English translation: "The Apprenticeship of a Mathematician" (1992), ISBN 0817626506
* "God exists since mathematics is consistent, and the Devil exists since we cannot prove it."
* Weil's Law of university hiring: "First rate people hire other first rate people. Second rate people hire third rate people. Third rate people hire fifth rate people."
Weil conjecturedisambiguation page
Weil conjecture on Tamagawa numbers
Weil group, Weil-Deligne group scheme
Weil reciprocity law
De Rham-Weil theorem
* [http://www.ams.org/notices/199904/index.html André Weil] : memorial articles in the Notices of AMS by
Armand Borel, Pierre Cartier, Komaravolu Chandrasekharan, Shiing-Shen Chern, and Shokichi Iyanaga
* [http://www.ams.org/images/weil-photo.gifImage of Weil]
* [http://www.ams.org/notices/200503/fea-weil.pdf A 1940 Letter of André Weil on Analogy in Mathematics]
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