André Weil (/veɪ/; French: [ɑ̃dʁe vɛj]; 6 May 1906 – 6 August 1998) was an influential French mathematician of the 20th century,^{[3]} known for his foundational work in number theory and algebraic geometry. He was a founding member and the de facto early leader of the mathematical Bourbaki group. The philosopher Simone Weil was his sister.^{[4]}^{[5]}
André Weil  

Born  6 May 1906 Paris, France 
Died  6 August 1998 (aged 92) Princeton, New Jersey, U.S. 
Alma mater  University of Paris École Normale Supérieure Aligarh Muslim University 
Known for  Contributions in number theory, algebraic geometry 
Awards 

Scientific career  
Fields  Mathematics 
Institutions  Aligarh Muslim University (1930–32) 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 
André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of AlsaceLorraine by the German Empire after the FrancoPrussian War in 1870–71. The famous philosopher Simone Weil was Weil's 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. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he taught himself Sanskrit in 1920.^{[6]}^{[7]} After teaching for one year in AixMarseille University, he taught for 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. His wife Éveline returned to France without him. Weil was mistakenly 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.^{[8]} 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 BonneNouvelle, 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, he met up with his family in Marseille, where he arrived by sea. He then went to ClermontFerrand, where he managed to join his wife Éveline, who had been living in Germanoccupied 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 didn't have to worry about being drafted, unlike his American students. But, he hated Lehigh very much for their heavy teaching workload and he swore that he would never talk about "Lehigh" any more. He quit the job at Lehigh, and then he moved to Brazil and taught at the Universidade de São Paulo from 1945 to 1947, where he worked with Oscar Zariski. 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,^{[9]} in 1954 in Amsterdam,^{[10]} and in 1978 in Helsinki.^{[11]} In 1979, Weil shared the second Wolf Prize in Mathematics with Jean Leray.
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).^{[12]} Mordell's theorem had an ad hoc proof;^{[13]} 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 zetafunctions of curves over finite fields,^{[14]} and his subsequent laying of proper foundations for algebraic geometry to support that result (from 1942 to 1946, most intensively). The socalled Weil conjectures were hugely influential from around 1950; these statements were later proved by Bernard Dwork,^{[15]} Alexander Grothendieck,^{[16]}^{[17]}^{[18]} Michael Artin, and finally by Pierre Deligne, who completed the most difficult step in 1973.^{[19]}^{[20]}^{[21]}^{[22]}^{[23]}
Weil introduced the adele ring^{[24]} 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).^{[25]} 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^{[26]} 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.^{[27]}
Other significant results were on Pontryagin duality and differential geometry.^{[28]} He introduced the concept of a uniform space in general topology, as a byproduct 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 socalled Weil representation, previously introduced in quantum mechanics by Irving Segal and 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.
He also wrote several books on the history of Number Theory. Weil was elected Foreign Member of the Royal Society (ForMemRS) in 1966.^{[1]}
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 that it came from the Norwegian alphabet, which he alone among the Bourbaki group was familiar with.^{[30]}
Indian (Hindu) thought had great influence on Weil.^{[31]} Although he was an agnostic,^{[32]} he respected religions.^{[33]}
Mathematical works:
Collected papers:
Memoir by his daughter:
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,...
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.
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