# André Weil : biography

Memoir by his daughter: *At Home with Andre and Simone Weil by Sylvie Weil, translated by Benjamin Ivry; ISBN 978-0-8101-2704-3, Northwestern University Press, 2010.*

## Beliefs

Indian (Hindu) thought had great influence on Weil.Borel, Armand. (see also) In his autobiography, he says that the only religious ideas that appealed to him were those to be found in Hindu philosophical thought. Although he was an agnostic, he respected religions.

## As expositor

Weil’s ideas made an important contribution to the writings and seminars of Bourbaki, before and after World War II.

He says on page 114 of his autobiography that he was responsible for the null set symbol (Ø) and it came from the Norwegian alphabet, with which he alone among the Bourbaki group was familiar.

## 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, of the Riemann hypothesis for zeta-functions of curves over finite fields, 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, Alexander Grothendieck, Michael Artin, and finally by 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 (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.

Other significant results were on Pontryagin duality and differential geometry. 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 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.

He also wrote several books on the history of Number Theory.

## 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."

## Notes

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