André Weil (; ; 6 May 1906 – 6 August 1998) was a French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, known for his foundational work in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

and

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

. He was a founding member and the ''de facto'' early leader of the mathematical

Bourbaki group Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook ...

. The

philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...

Simone Weil was his sister. The writer

Sylvie Weil Sylvie Weil (born 1942) is a French professor and writer. She is known for her novels for children and her writing about her prominent intellectual family, which includes André Weil and Simone Weil. Biography Weil was born in the United States ...

is his daughter.

Life

André Weil was born in

Paris Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. Si ...

agnostic Agnosticism is the view or belief that the existence of God, of the divine or the supernatural is unknown or unknowable. (page 56 in 1967 edition) Another definition provided is the view that "human reason is incapable of providing sufficien ...

Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the

German Empire The German Empire (),Herbert Tuttle wrote in September 1881 that the term "Reich" does not literally connote an empire as has been commonly assumed by English-speaking people. The term literally denotes an empire – particularly a hereditary ...

after the Franco-Prussian War in 1870–71. Simone Weil, who would later become a famous philosopher, was Weil's younger sister and only sibling. He studied in Paris,

Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus ( legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...

and

Göttingen Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. General information The ori ...

and received his

doctorate A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism ''l ...

in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at

Aligarh Muslim University Aligarh Muslim University (abbreviated as AMU) is a Public University, public Central University (India), central university in Aligarh, Uttar Pradesh, India, which was originally established by Sir Syed Ahmad Khan as the Muhammadan Anglo-Orie ...

in India. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in

Hinduism Hinduism () is an Indian religion or '' dharma'', a religious and universal order or way of life by which followers abide. As a religion, it is the world's third-largest, with over 1.2–1.35 billion followers, or 15–16% of the global p ...

and

Sanskrit literature Sanskrit literature broadly comprises all literature in the Sanskrit language. This includes texts composed in the earliest attested descendant of the Proto-Indo-Aryan language known as Vedic Sanskrit, texts in Classical Sanskrit as well as ...

: he had taught himself Sanskrit in 1920.Amir D. Acze
''The Artist and the Mathematician,''
Basic Books, 2009 pp.17ff.,p.25. After teaching for one year at Aix-Marseille University, he taught for six years at

University of Strasbourg The University of Strasbourg (french: Université de Strasbourg, Unistra) is a public research university located in Strasbourg, Alsace, France, with over 52,000 students and 3,300 researchers. The French university traces its history to the ea ...

. He married Éveline de Possel (née Éveline Gillet) in 1937. Weil was in

Finland Finland ( fi, Suomi ; sv, Finland ), officially the Republic of Finland (; ), is a Nordic country in Northern Europe. It shares land borders with Sweden to the northwest, Norway to the north, and Russia to the east, with the Gulf of Bot ...

when

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

broke out; he had been traveling in Scandinavia since April 1939. His wife Éveline returned to France without him. Weil was arrested in Finland at the outbreak of the

Winter War The Winter War,, sv, Vinterkriget, rus, Зи́мняя война́, r=Zimnyaya voyna. The names Soviet–Finnish War 1939–1940 (russian: link=no, Сове́тско-финская война́ 1939–1940) and Soviet–Finland War 1 ...

on suspicion of spying; however, accounts of his life having been in danger were shown to be exaggerated. Weil returned to France via Sweden and the United Kingdom, and was detained at

Le Havre Le Havre (, ; nrf, Lé Hâvre ) is a port city in the Seine-Maritime department in the Normandy region of northern France. It is situated on the right bank of the estuary of the river Seine on the Channel southwest of the Pays de Caux, very ...

in January 1940. He was charged with failure to report for duty, and was imprisoned in Le Havre and then

Rouen Rouen (, ; or ) is a city on the River Seine in northern France. It is the prefecture of the region of Normandy and the department of Seine-Maritime. Formerly one of the largest and most prosperous cities of medieval Europe, the population ...

. It was in the military prison in Bonne-Nouvelle, 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 Cherbourg (; , , ), nrf, Chèrbourg, ) is a former commune and subprefecture located at the northern end of the Cotentin peninsula in the northwestern French department of Manche. It was merged into the commune of Cherbourg-Octeville on 28 Febr ...

. After the

fall of France The Battle of France (french: bataille de France) (10 May – 25 June 1940), also known as the Western Campaign ('), the French Campaign (german: Frankreichfeldzug, ) and the Fall of France, was the German invasion of France during the Second World ...

in June 1940, he met up with his family in

Marseille Marseille ( , , ; also spelled in English as Marseilles; oc, Marselha ) is the prefecture of the French department of Bouches-du-Rhône and capital of the Provence-Alpes-Côte d'Azur region. Situated in the camargue region of southern Fra ...

, where he arrived by sea. He then went to

Clermont-Ferrand Clermont-Ferrand (, ; ; oc, label= Auvergnat, Clarmont-Ferrand or Clharmou ; la, Augustonemetum) is a city and commune of France, in the Auvergne-Rhône-Alpes region, with a population of 146,734 (2018). Its metropolitan area (''aire d'attrac ...

, where he managed to join his wife Éveline, who had been living in German-occupied 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 The Rockefeller Foundation is an American private foundation and philanthropy, philanthropic medical research and arts funding organization based at 420 Fifth Avenue, New York City. The second-oldest major philanthropic institution in America, aft ...

and the Guggenheim Foundation. For two years, he taught undergraduate mathematics at Lehigh University, where he was unappreciated, overworked and poorly paid, although he did not have to worry about being drafted, unlike his American students. He quit the job at Lehigh and moved to Brazil, where he taught at the

Universidade de São Paulo The Universiade is an international multi-sport event, organized for university athletes by the International University Sports Federation (FISU). The name is a portmanteau of the words "University" and "Olympiad". The Universiade is referred t ...

from 1945 to 1947, working with Oscar Zariski. Weil and his wife had two daughters, Sylvie (born in 1942) and Nicolette (born in 1946). He then returned to the United States and taught at the

University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...

from 1947 to 1958, before moving to the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...

, where he would spend the remainder of his career. He was a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts, in 1954 in Amsterdam, and in 1978 in Helsinki. Weil was elected Foreign Member of the Royal Society in 1966. In 1979, he shared the second Wolf Prize in Mathematics with Jean Leray.

Work

Weil made substantial contributions in a number of areas, the most important being his discovery of profound connections between

and

. This began in his doctoral work leading to the

Mordell–Weil theorem In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of k-rational point, ''K''-rational points of A is a finitely-generated abelian group, called the Mordell–Weil group. The case ...

(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 In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold f ...

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 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 Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Pierre Deligne, who completed the most difficult step in 1973. Weil introduced the adele ring in the late 1930s, following

Claude Chevalley Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a fou ...

'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). His 'matrix divisor' (

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...

''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 In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset ...

theory. He picked up another credited ''Weil conjecture'', around 1967, which later under pressure from

Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...

(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 Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

. He introduced the concept of a uniform space in general topology, as a by-product of his collaboration with

Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook i ...

(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 also chose the symbol

∅ In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

, derived from the letter Ø in the Norwegian alphabet (which he alone among the Bourbaki group was familiar with), to represent the

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

. Weil also made a well-known contribution in Riemannian geometry in his very first paper in 1926, when he showed that the classical

isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...

holds on non-positively curved surfaces. This established the 2-dimensional case of what later became known as the Cartan–Hadamard conjecture. He discovered that the so-called Weil representation, previously introduced in

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

Irving Segal Irving Ezra Segal (1918–1998) was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for h ...

and

David Shale David Winston Howard Shale (22 March 1932, New Zealand – 7 January 2016) was a New Zealand-American mathematician, specializing in the mathematical foundations of quantum physics. He is known as one of the namesakes of the Segal–Shale-Weil re ...

, gave a contemporary framework for understanding the classical theory of

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...

s. This was also a beginning of a substantial development by others, connecting representation theory and

theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...

s. Weil was a member of both the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...

and the

American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...

As expositor

Weil's ideas made an important contribution to the writings and seminars of Bourbaki, before and after

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposi ...

. He also wrote several books on the history of number theory.

Beliefs

Indian (Hindu) thought had great influence on Weil. He was an agnostic, and he respected religions.

Legacy

Asteroid 289085 Andreweil, discovered by astronomers at the

Saint-Sulpice Observatory This is a list of observatory codes (IAU codes or MPC codes) published by the Minor Planet Center The Minor Planet Center (MPC) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical ...

in 2004, was named in his memory. The official was published by the

Minor Planet Center The Minor Planet Center (MPC) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory. Function T ...

on 14 February 2014 ().

Books

Mathematical works: * ''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) * * ''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 * * ''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) * ''Number Theory for Beginners'' (1979) with Maxwell Rosenlicht * ''Adeles and Algebraic Groups'' (1982)

(1984) Collected papers: * ''Œuvres Scientifiques, Collected Works, three volumes'' (1979) * * *

Autobiography An autobiography, sometimes informally called an autobio, is a self-written account of one's own life. It is a form of biography. Definition The word "autobiography" was first used deprecatingly by William Taylor in 1797 in the English peri ...

: * French: ''Souvenirs d'Apprentissage'' (1991)
Review in English
by J. E. Cremona. * English translation

(1992),
Review
by Veeravalli S. Varadarajan
Review
by

Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...

Memoir by his daughter:
''At Home with André and Simone Weil''
by Sylvie Weil, translated by

Benjamin Ivry Benjamin Ivry is an American writer on the arts, broadcaster and translator. Ivry is author of biographies of Francis Poulenc, Arthur Rimbaud, and Maurice Ravel, as well as a poetry collection, ''Paradise for the Portuguese Queen''. The latter co ...

; ,

Northwestern University Press Northwestern University Press is an American publishing house affiliated with Northwestern University in Evanston, Illinois. It publishes 70 new titles each year in the areas of continental philosophy, poetry, Slavic and German literary criticism ...

, 2010.

References

External links

André Weil
by A. Borel, Bull. AMS 46 (2009), 661–666.

memorial articles in the '' Notices of the AMS'' by

Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in ...

, Pierre Cartier, Komaravolu Chandrasekharan, Shiing-Shen Chern, and Shokichi Iyanaga
Image of Weil

A 1940 Letter of André Weil on Analogy in Mathematics
* *
Artless innocents and ivory-tower sophisticates: Some personalities on the Indian mathematical scene
– M. S. Raghunathan * * La vie et l'oeuvre d'André Weil, by J-P. Serre, L'Ens. Math. 45 (1999),5–16. * Correspondence entre Henri Cartan et André Weil (1928–1991), par Michèle Audin, Doc. Math. 6, Soc. Math. France, 2011. {{DEFAULTSORT:Weil, Andre 1906 births 1998 deaths 20th-century French mathematicians Jewish French scientists French historians of mathematics Jewish agnostics French agnostics French people of Jewish descent Institute for Advanced Study faculty Aligarh Muslim University faculty Arithmetic geometers École Normale Supérieure alumni Nicolas Bourbaki Members of the French Academy of Sciences Kyoto laureates in Basic Sciences Wolf Prize in Mathematics laureates Aligarh Muslim University alumni University of São Paulo faculty Foreign Members of the Royal Society Foreign associates of the National Academy of Sciences Scientists from Paris Lycée Saint-Louis alumni 20th-century French historians Members of the American Philosophical Society French expatriates in Brazil French expatriates in the United States