André Weil

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André Weil (1956)

André Weil (born May 6, 1906 in Paris , † August 6, 1998 in Princeton ) was a French mathematician .


André Weil grew up as the son of a Jewish doctor in Paris and during the First World War in southern France. The philosopher Simone Weil was his sister. The family lived in Alsace, but moved from there after it was annexed by the German Empire in 1871. Weil is also distantly related to Albert Schweitzer . At the age of 16, he enrolled at the École normal supérieure . After stays abroad in Rome and Göttingen , he received his doctorate in 1928 at the age of 22 under Jacques Hadamard with a thesis on Diophantine equations . From 1930 to 1932 he lived in India ( Aligarh Muslim University ), then in Marseille and for six years in Strasbourg.

Together with some former fellow students, he founded the Bourbaki circle in the early 1930s - at that time he was a professor in Strasbourg ; the naming of the group should come from him. In 1937 he married Eveline, previously married to de Possel .

When the Second World War broke out, Weil fled to Finland before his military service, where Rolf Nevanlinna visited . Since Finland was in the winter war with the Soviet Union , letters of the mathematician Lev Pontryagin in Russian that were found near Weil led to his arrest as a spy. In his autobiography, Weil even describes that he should have been shot; Nevanlinna managed to have him expelled instead. In France, Weil was imprisoned for deserting in Rouen, but escaped trial by volunteering. In 1941 he and his wife fled to the USA.

In the United States he lived on grants from the Guggenheim and Rockefeller Foundations . After what he felt was a very frustrating teaching position at "Pennsylvania engineering schools" ( Haverford College , Swarthmore College ) and an interlude in São Paulo from 1945 to 1947 (where he met Oscar Zariski ), he moved to Chicago in 1947 and then to the Institute for Advanced in 1958 Study at Princeton . There he retired in 1976, but continued to work.

Weil was known for his sharp-talkedness and contentiousness. In his book on the history of the IAS, Ed Regis reports on the intrigues of Weil and other institute members against the head of the Institute for Advanced Study, the economist Carl Kaysen (about whom Weil said contemptuously: I think he wrote his dissertation on a shoe factory ). The opposition escalated into a public scandal in 1973 that made the headlines of the New York Times when Kaysen, against the vote of the majority of the permanent members, made the sociologist Robert N. Bellah a permanent member of the IAS; the mathematicians Weil, Armand Borel , Deane Montgomery and other members were strongly against it, but Kaysen received support from other members and initially remained director, but then left voluntarily two years later by the intrigues in 1976. Weil also tried, as he told Regis , his own retirement at the institute, which fell on the same day as Kaysen's departure, to extend at least 24 hours, as he said, to enjoy at least one “Kaysen-free” day at the institute. He is also well known for his harsh condemnation of Michael S. Mahoney's Fermat biography.

In 1950 he gave a plenary lecture at the ICM in Cambridge ( Number Theory and Algebraic Geometry ), in 1954 at the Amsterdam ( Abstract versus Classical Algebraic Geometry ) and in 1978 at the ICM in Helsinki ( History of Mathematics: Why and How ). In 1980 he received the Leroy P. Steele Prize from the American Mathematical Society . Since 1962 he was a corresponding member of the Bavarian Academy of Sciences . In 1966 he was elected a foreign member of the Royal Society , in 1977 in the National Academy of Sciences , in 1982 in the Académie des sciences and in 1995 in the American Philosophical Society .

Weil was seen as a promising candidate for the Fields Medal in 1950, but was then outmaneuvered by committee chairman Harald Bohr , who favored Laurent Schwartz , with the argument that Weil was already one of the most respected mathematicians and that the medal should honor younger mathematicians (Weil was 43 years old at the time old) and not the greatest math genius. But there were still reservations until Bohr finally prevailed, who allied himself with Marston Morse , who favored Atle Selberg .


André Weil was one of the outstanding mathematicians of the 20th century. The focus of his work was on the areas of algebraic geometry and number theory , between which he found surprising connections.

In his dissertation in 1928 he proved the Mordell-Weil theorem . It says that the group of rational points is finite-generated on an Abelian variety (which means something like defined by algebraic equations and given a group structure). Louis Mordell had already proven the special case of elliptical curves . The group structure in this special case goes back to Henri Poincaré and his tangent construction of rational points on elliptic curves. Weil transferred the idea of ​​Fermat's "infinite descent" proof in the theory of Diophantine equations with the help of the introduction of "height functions", which made it possible to measure the "size" of rational points on algebraic curves.

Another goal of Weil in the 1930s was to prove the Riemann Hypothesis for zeta functions on Abelian varieties . Helmut Hasse had already dealt with the special case of elliptical curves . Weil succeeded in proving this in 1940 while he was in prison in France. He spent the rest of the 1940s putting algebraic geometry on a strict algebraic basis in order to back up his proofs (books Foundations of algebraic geometry 1946 et al.).

In 1945 he found a deep connection between the zeta function of an algebraic manifold over finite fields and the topology ( Betti numbers, among others) of this algebraic manifold. The term zeta function of an algebraic variety has to be imagined as a kind of counting function for the number of points of this curve in the body. He formulated this in his famous “ because conjectures ”. They say u. a. that the zeta function is a rational function (quotient of polynomials), that the degrees of the polynomials are equal to the Betti numbers of the underlying manifold, the zeta function satisfies a functional equation and that the zeros have the real part ½ (“ Riemann conjecture ”). The rationality was proven by Dwork with "elementary" p-adic methods. For the last, the “Riemann Conjecture”, Pierre Deligne needed the entire huge building of algebraic geometry that the Grothendieck School had built in 1974 . Weil himself had proven the special case of curves. In his own words, Weil found the inspiration for the whole theory in the study of Gauss' works (Gauss sums). Weil goes into this in La cyclotomie jadis et naguère (The division of circles then and now), but the connection is also shown in A classical introduction to modern number theory by Rosen and Ireland .

Another conjecture named after him is the Taniyama-Shimura-Weil conjecture , which was proven in 1999. It says that elliptic curves over the rational numbers are parameterized by modular functions . A special case of this conjecture, which implied the correctness of the Fermat conjecture , was proven in 1995 by Andrew Wiles and Richard Taylor . Under pressure from the no less controversial Serge Lang , the "because" was increasingly relativized in the presumption. Weil himself did not come up with the presumption first, but did a lot of work in the 1960s to support it.

In his book Basic number theory from 1967 he followed his own original approach using Claude Chevalley's "Ideles" and the "Adeles" he developed from them, the integration via topological groups and group cohomology in the form of "central simple algebras" .

He also introduced harmonic analysis on topological groups (book of the same name, 1940) and wrote a book on Kahler manifolds in 1958 . The Weil representations are important in mathematical formulations of quantum mechanics and were introduced by Weil as a representation-theoretical interpretation of the theory of the theta function (in relation to symplectic groups).

With Carl B. Allendoerfer in 1943 he generalized Gauss-Bonnet's theorem to higher dimensions.

Thanks to his classical education (he was a passionate collector of antiquarian books, spoke the ancient languages ​​fluently and studied Sanskrit in Paris) he was also interested in the history of mathematics, especially Pierre de Fermat . A large number of books and articles (as well as sharp reviews) bear witness to this. He also published the works of Ernst Eduard Kummer .


  • Oeuvres Scientifiques- Collected papers , 3 volumes, Springer Verlag, 1979 (with his commentary)
  • Apprenticeship and wandering years of a mathematician , Birkhäuser 1993 (Original Souvenir d'apprentissage , Birkhäuser Verlag, Basel, 1991, 201 pp, ISBN 3-7643-2500-3 ) (autobiography, only available until the end of 1947)
  • Michèle Audin (editor) Correspondance entre Henri Cartan et André Weil (1928–1991) , Documents Mathématiques 6, Société Mathématique de France, 2011.
  • Numbers of solutions of equations in finite fields , Bulletin American Mathematical Society, Vol. 55, 1949, pp. 497-508
  • Basic number theory , Springer Verlag 1967, 1995
  • Elliptic functions according to Kronecker and Eisenstein , Springer Verlag, Results of Mathematics and their Frontier Areas, Volume 88, 1976
  • Number theory - a walk through the history of Hammurabi to Legendre , Birkhäuser 1992 (first English 1984)
  • Two lectures on number theory - past and present , L Enseignement Mathematique 1974
  • La cyclotomie jadis et naguère , Bourbaki Seminar 1974, online here Weil: La cyclotomie jadis et naguère
  • Dirichlet series and automorphic forms , Springer 1971
  • Courbes algebriques et varietes abeliennes , Hermann 1971
  • Adeles and Algebraic Groups , Birkhäuser 1982
  • Number theory for beginners , Springer 1979 (70 pages, with the participation of Maxwell Rosenlicht )
  • Arithmétique et géométrie sur les variétés algébriques , Hermann 1935
  • L'intégration dans les groupes topologiques et ses applications , 1941, 2nd edition, Hermann 1951
  • Foundations of algebraic geometry , American Mathematical Society (AMS), 1947, 1962
  • Introduction à l'étude des variétés kählériennes , Hermann 1958
  • L'arithmétique sur les courbes algébriques , dissertation 1928


  • André Weil: Apprenticeship and wandering years of a mathematician , Birkhäuser 1993
  • Freitag, Kiehl: Etale cohomology and the Weil conjecture , Springer Verlag 1988 (in appendix Jean Dieudonné on the story)
  • Osmo Pekonen: L'affaire Weil à Helsinki en 1939 , Gazette des mathématiciens 52 (avril 1992), pp. 13-20. With an afterword by André Weil (Weil wrote in his autobiography that he was arrested as a spy there, that he was threatened with shooting and that he was only released after Rolf Nevanlinna interceded - the facts are much less dramatic according to Pekonen).
  • Pierre Cartier Farewell to a friend- André Weil (1906–1998) , DMV Mitteilungen 1999, No. 3, p. 9
  • Jean-Pierre Serre: André Weil , Biographical Memoirs Fellows Royal Society, Volume 45, 1999, pp. 519-529

See also

Web links

Individual evidence

  1. ^ Regis, Who got Einstein's office. Eccentricity and Genius at the Institute for Advanced Study , Basic Books 1987, pp. 205ff
  2. ^ Ed Regis, Who got Einstein's office, p. 204.
  3. a b Regis, Who got Einstein's office, p. 206
  4. Martina Schneider: Contextualizing Unguru's 1975 attack on the historiography of ancient greek mathematics , in: Volker Remmert, Martina Schneider, Henrik Kragh Sörensen (eds.): Historiography of Mathematics in the 19th and 20th centuries , Birkhäuser 2016 p. 259.
  5. Martina Schneider: Contextualizing Unguru's 1975 attack on the historiography of ancient greek mathematics , p. 260
  6. ^ André Weil obituary in the 1999 yearbook of the Bavarian Academy of Sciences (PDF file)
  7. ^ Entry on Weil, André (1906–1998) in the archive of the Royal Society , London
  8. Member History: André Weil. American Philosophcal Society, accessed July 21, 2018 .
  9. Michael Barany, The Fields Medal should return to its roots , Nature, January 12, 2018