André Weil

André Weil (6 May 1906 – 6 August 1998) (French pronunciation: [ɑ̃dʁe vɛj]) was an influential mathematician 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 theory and algebraic geometry. He was a founding member and the de facto early leader of the influential Bourbaki group. The philosopher Simone Weil was his sister.

Life

Born in Paris to Alsatian agnostic Jewish parents who fled the annexation of Alsace-Lorraine to Germany, Weil studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, he befriended Carl Ludwig Siegel. He spent two academic years at Aligarh Muslim University from 1930. Hinduism and Sanskrit literature were his life-long interests. After one year in Marseille, he taught six years in Strasbourg. He married Éveline in 1937.

Weil was in Finland when World War II broke out; he had been traveling in Scandinavia since April 1939. Éveline returned to France without him. Weil was mistakenly arrested in Finland at the outbreak of the Winter War suspected of spying; however, accounts of his life having been at danger have been shown to be exaggerated.[1] 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 he did the work that made his reputation. He was tried on 3 May 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 Éveline, who had been in German-occupied France.

In January 1941, Weil and his family sailed from Marseille to New York. He spent the war in the United States, where he was supported by the Rockefeller Foundation and 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 Chicago from 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.

Work

He made substantial contributions in many 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). Mordell's theorem had an ad hoc proof; 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 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 hypothesis for 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 conjectures were 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 ring in the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann–Roch theorem with them (a version appeared in his Basic Number Theory in 1967). 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 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 became known as the Taniyama–Shimura conjecture based 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 duality and differential geometry. He introduced the concept of uniform space in general topology. 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 discovered that the so-called Weil representation, previously introduced in quantum mechanics by Irving Segal and 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 theory and theta-functions.

As expositor

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, an exceptional situation in mathematics. (In one famous case, the influence was possibly negative: Alexander Grothendieck is 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 empty set (q.v.).

Beliefs

Indian (Hindu) thought had great influence on Weil. In his autobiography, he says that the only religious ideas that appealed to him were those to be found in Hindu philosophical thought. During the second World War, Weil refused to do military service and cited the Bhagavad Gita to justify his stand: he said his true dharma was the pursuit of mathematics and that was what he should be doing, not assisting in the war effort, however just the cause.

Books

* 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)

His autobiography:

* French: Souvenirs d’Apprentissage (1991) ISBN 3764325003. Review in English by J. E. Cremona.
* English translation: The Apprenticeship of a Mathematician (1992), ISBN 0817626506

Quotations

* "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."