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André Weil

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André Weil
Born(1906-05-06)6 May 1906
Paris,France
Died6 August 1998(1998-08-06)(aged 92)
Princeton, New Jersey,U.S.
Education
Known for
List
Awards
Scientific career
FieldsMathematics
Institutions
Doctoral advisor
Doctoral students

André Weil(/ˈveɪ/;French:[ɑ̃dʁevɛj];6 May 1906 – 6 August 1998) was a Frenchmathematician,known for his foundational work innumber theoryandalgebraic geometry.[3]He was one of the most influential mathematicians of the twentieth century. His influence is due both to his original contributions to a remarkably broad spectrum of mathematical theories, and to the mark he left on mathematical practice and style, through some of his own works as well as through theBourbaki group,of which he was one of the principal founders.

Life[edit]

André Weil was born inParistoagnosticAlsatian Jewishparents who fled the annexation ofAlsace-Lorraineby theGerman Empireafter theFranco-Prussian Warin 1870–71.Simone Weil,who would later become a famous philosopher, was Weil's younger sister and only sibling. He studied in Paris,RomeandGöttingenand received hisdoctoratein 1928. While in Germany, Weil befriendedCarl Ludwig Siegel.Starting in 1930, he spent two academic years atAligarh Muslim Universityin India. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature,HinduismandSanskrit literature:he had taught himself Sanskrit in 1920.[4][5]After teaching for one year atAix-Marseille University,he taught for six years atUniversity of Strasbourg.He married Éveline de Possel (née Éveline Gillet) in 1937.[6]

Weil was inFinlandwhenWorld War IIbroke 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 theWinter Waron suspicion of spying; however, accounts of his life having been in danger were shown to be exaggerated.[7]Weil returned to France via Sweden and the United Kingdom, and was detained atLe Havrein January 1940. He was charged withfailure to report for duty,and was imprisoned in Le Havre and thenRouen.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 inCherbourg.After thefall of Francein June 1940, he met up with his family inMarseille,where he arrived by sea. He then went toClermont-Ferrand,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 theRockefeller Foundationand theGuggenheim Foundation.For two years, he taught undergraduate mathematics atLehigh 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 theUniversidade de São Paulofrom 1945 to 1947, working withOscar Zariski.Weil and his wife had two daughters,Sylvie(born in 1942) and Nicolette (born in 1946).[6]

He then returned to the United States and taught at theUniversity of Chicagofrom 1947 to 1958, before moving to theInstitute for Advanced Study,where he would spend the remainder of his career. He was a Plenary Speaker at theICMin 1950 in Cambridge, Massachusetts,[8]in 1954 in Amsterdam,[9]and in 1978 in Helsinki.[10]Weil was electedForeign Member of the Royal Society in 1966.[1]In 1979, he shared the secondWolf Prize in MathematicswithJean Leray.

Work[edit]

Weil made substantial contributions in a number of areas, the most important being his discovery of profound connections betweenalgebraic geometryandnumber theory.This began in his doctoral work leading to theMordell–Weil theorem(1928, and shortly applied inSiegel's theorem on integral points).[11]Mordell's theoremhad anad hocproof;[12]Weil began the separation of theinfinite descentargument into two types of structural approach, by means ofheight functionsfor sizing rational points, and by means ofGalois 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 theRiemann hypothesis for zeta-functionsof curves over finite fields,[13]and his subsequent laying of properfoundations for algebraic geometryto support that result (from 1942 to 1946, most intensively). The so-calledWeil conjectureswere hugely influential from around 1950; these statements were later proved byBernard Dwork,[14]Alexander Grothendieck,[15][16][17]Michael Artin,and finally byPierre Deligne,who completed the most difficult step in 1973.[18][19][20][21][22]

Weil introduced theadele ring[23]in the late 1930s, followingClaude Chevalley's lead with theideles,and gave a proof of theRiemann–Roch theoremwith them (a version appeared in hisBasic Number Theoryin 1967).[24]His 'matrix divisor' (vector bundleavant la lettre) Riemann–Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles. TheWeil conjecture on Tamagawa numbers[25]proved resistant for many years. Eventually the adelic approach became basic inautomorphic representationtheory. He picked up another creditedWeil conjecture,around 1967, which later under pressure fromSerge Lang(resp. of Serre) became known as theTaniyama–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.[26]

Other significant results were onPontryagin dualityanddifferential geometry.[27]He introduced the concept of auniform spaceingeneral topology,as a by-product of his collaboration withNicolas Bourbaki(of which he was a Founding Father). His work onsheaf theoryhardly appears in his published papers, but correspondence withHenri Cartanin the late 1940s, and reprinted in his collected papers, proved most influential. He also chose the symbol∅,derived from the letterØin theNorwegian Alpha bet(which he alone among the Bourbaki group was familiar with), to represent theempty set.[28]

Weil also made a well-known contribution inRiemannian geometryin his very first paper in 1926, when he showed that the classicalisoperimetric inequalityholds on non-positively curved surfaces. This established the 2-dimensional case of what later became known as theCartan–Hadamard conjecture.

He discovered that the so-calledWeil representation,previously introduced inquantum mechanicsbyIrving SegalandDavid Shale,gave a contemporary framework for understanding the classical theory ofquadratic forms.[29]This was also a beginning of a substantial development by others, connectingrepresentation theoryandtheta functions.

Weil was a member of both theNational Academy of Sciences[30]and theAmerican Philosophical Society.[31]

As expositor[edit]

Weil's ideas made an important contribution to the writings and seminars ofBourbaki,before and afterWorld War II.He also wrote several books on the history of number theory.

Beliefs[edit]

Hindu thoughthad great influence on Weil.[32]He was an agnostic,[33]and he respected religions.[34]

Legacy[edit]

Asteroid289085 Andreweil,discovered by astronomers at theSaint-Sulpice Observatoryin 2004, was named in his memory.[35]The officialnaming citationwas published by theMinor Planet Centeron 14 February 2014 (M.P.C.87143).[36]

Books[edit]

Mathematical works:

  • Arithmétique et géométrie sur les variétés algébriques(1935)[37]
  • Sur les espaces à structure uniforme et sur la topologie générale(1937)[38]
  • L'intégration dans les groupes topologiques et ses applications(1940)
  • Weil, André(1946),Foundations of Algebraic Geometry,American Mathematical Society Colloquium Publications, vol. 29, Providence, R.I.:American Mathematical Society,ISBN978-0-8218-1029-3,MR0023093[39]
  • Sur les courbes algébriques et les variétés qui s'en déduisent(1948)
  • Variétés abéliennes et courbes algébriques(1948)[40]
  • Introduction à l'étude des variétés kählériennes(1958)
  • Discontinuous subgroups of classical groups(1958) Chicago lecture notes
  • Weil, André (1967),Basic number theory.,Die Grundlehren der mathematischen Wissenschaften, vol. 144, Springer-Verlag New York, Inc., New York,ISBN3-540-58655-5,MR0234930[41]
  • Dirichlet Series and Automorphic Forms, Lezioni Fermiane(1971) Lecture Notes in Mathematics, vol. 189[42]
  • Essais historiques sur la théorie des nombres(1975)
  • Elliptic Functions According to Eisenstein and Kronecker(1976)[43]
  • Number Theory for Beginners(1979) with Maxwell Rosenlicht[44]
  • Adeles and Algebraic Groups(1982)[45]
  • Number Theory: An Approach Through History From Hammurapi to Legendre(1984)[46]

Collected papers:

Autobiography:

Memoir by his daughter:

See also[edit]

References[edit]

  1. ^abSerre, J.-P.(1999)."Andre Weil. 6 May 1906 – 6 August 1998: Elected For.Mem.R.S. 1966".Biographical Memoirs of Fellows of the Royal Society.45:519.doi:10.1098/rsbm.1999.0034.
  2. ^André Weilat theMathematics Genealogy Project
  3. ^Horgan, J(1994). "Profile: Andre Weil – The Last Universal Mathematician".Scientific American.270(6): 33–34.Bibcode:1994SciAm.270f..33H.doi:10.1038/scientificamerican0694-33.
  4. ^Amir D. Aczel,The Artist and the Mathematician,Basic Books, 2009 pp. 17ff., p. 25.
  5. ^Borel, Armand
  6. ^abYpsilantis, Olivier (31 March 2017)."En lisant" Chez les Weil. André et Simone "".Retrieved26 April2020.
  7. ^Osmo Pekonen:L'affaire Weil à Helsinki en 1939,Gazette des mathématiciens 52 (avril 1992), pp. 13–20. With an afterword by André Weil.
  8. ^Weil, André."Number theory and algebraic geometry."Archived30 August 2017 at theWayback MachineIn Proc. Intern. Math. Congres., Cambridge, Mass., vol. 2, pp. 90–100. 1950.
  9. ^Weil, A."Abstract versus classical algebraic geometry"(PDF).In:Proceedings of International Congress of Mathematicians, 1954, Amsterdam.Vol. 3. pp. 550–558.Archived(PDF)from the original on 9 October 2022.
  10. ^Weil, A."History of mathematics: How and why"(PDF).In:Proceedings of International Congress of Mathematicians, (Helsinki, 1978).Vol. 1. pp. 227–236.Archived(PDF)from the original on 9 October 2022.
  11. ^A. Weil,L'arithmétique sur les courbes algébriques,Acta Math 52, (1929) p. 281–315, reprinted in vol 1 of his collected papersISBN0-387-90330-5 .
  12. ^L.J. Mordell,On the rational solutions of the indeterminate equations of the third and fourth degrees,Proc Cam. Phil. Soc. 21, (1922) p. 179
  13. ^Weil, André(1949), "Numbers of solutions of equations in finite fields",Bulletin of the American Mathematical Society,55(5): 497–508,doi:10.1090/S0002-9904-1949-09219-4,ISSN0002-9904,MR0029393Reprinted in Oeuvres Scientifiques/Collected Papers by André WeilISBN0-387-90330-5
  14. ^Dwork, Bernard(1960), "On the rationality of the zeta function of an algebraic variety",American Journal of Mathematics,82(3), American Journal of Mathematics, Vol. 82, No. 3: 631–648,doi:10.2307/2372974,ISSN0002-9327,JSTOR2372974,MR0140494
  15. ^Grothendieck, Alexander(1960),"The cohomology theory of abstract algebraic varieties",Proc. Internat. Congress Math. (Edinburgh, 1958),Cambridge University Press,pp. 103–118,MR0130879
  16. ^Grothendieck, Alexander(1995) [1965],"Formule de Lefschetz et rationalité des fonctions L",Séminaire Bourbaki,vol. 9, Paris:Société Mathématique de France,pp. 41–55,MR1608788
  17. ^Grothendieck, Alexander(1972),Groupes de monodromie en géométrie algébrique, I: Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I),Lecture Notes in Mathematics, vol. 288, Springer-Verlag,doi:10.1007/BFb0068688,ISBN978-3-540-05987-5,MR0354656
  18. ^Deligne, Pierre(1971),"Formes modulaires et représentations l-adiques",Séminaire Bourbaki vol. 1968/69 Exposés 347–363,Lecture Notes in Mathematics, vol. 179, Berlin, New York:Springer-Verlag,doi:10.1007/BFb0058801,ISBN978-3-540-05356-9
  19. ^Deligne, Pierre(1974),"La conjecture de Weil. I",Publications Mathématiques de l'IHÉS,43(43): 273–307,doi:10.1007/BF02684373,ISSN1618-1913,MR0340258,S2CID123139343
  20. ^Deligne, Pierre,ed. (1977),Cohomologie Etale,Lecture Notes in Mathematics (in French), vol. 569, Berlin:Springer-Verlag,doi:10.1007/BFb0091516,ISBN978-0-387-08066-6,archived fromthe originalon 15 May 2009
  21. ^Deligne, Pierre(1980),"La conjecture de Weil. II",Publications Mathématiques de l'IHÉS,52(52): 137–252,doi:10.1007/BF02684780,ISSN1618-1913,MR0601520,S2CID189769469
  22. ^Deligne, Pierre;Katz, Nicholas(1973),Groupes de monodromie en géométrie algébrique. II,Lecture Notes in Mathematics, Vol. 340, vol. 340, Berlin, New York:Springer-Verlag,doi:10.1007/BFb0060505,ISBN978-3-540-06433-6,MR0354657
  23. ^A. Weil,Adeles and algebraic groups,Birkhauser, Boston, 1982
  24. ^Weil, André (1967),Basic number theory.,Die Grundlehren der mathematischen Wissenschaften, vol. 144, Springer-Verlag New York, Inc., New York,ISBN3-540-58655-5,MR0234930
  25. ^Weil, André(1959),Exp. No. 186, Adèles et groupes algébriques,Séminaire Bourbaki, vol. 5, pp. 249–257
  26. ^Lang, S. "Some History of the Shimura-Taniyama Conjecture." Not. Amer. Math. Soc. 42, 1301–1307, 1995
  27. ^Borel, A. (1999)."André Weil and Algebraic Topology"(PDF).Notices of the AMS.46(4): 422–427.Archived(PDF)from the original on 9 October 2022.
  28. ^Miller, Jeff (1 September 2010)."Earliest Uses of Symbols of Set Theory and Logic".Jeff Miller Web Pages.Retrieved21 September2011.
  29. ^Weil, A. (1964)."Sur certains groupes d'opérateurs unitaires".Acta Math.(in French).111:143–211.doi:10.1007/BF02391012.
  30. ^"Andre Weil".nasonline.org.Retrieved20 December2021.
  31. ^"APS Member History".search.amphilsoc.org.Retrieved20 December2021.
  32. ^Borel, Armand.[1](see also)[2]
  33. ^Paul Betz; Mark Christopher Carnes,American Council of Learned Societies(2002).American National Biography: Supplement, Volume 1.Oxford University Press.p. 676.ISBN978-0-19-515063-6.Although as a lifelong agnostic he may have been somewhat bemused by Simone Weil's preoccupations withChristian mysticism,he remained a vigilant guardian of her memory,...
  34. ^I. Grattan-Guinness(2004). I. Grattan-Guinness, Bhuri Singh Yadav (ed.).History of the Mathematical Sciences.Hindustan Book Agency. p. 63.ISBN978-81-85931-45-6.Like in mathematics he would go directly to the teaching of the Masters. He readVivekanandaand was deeply impressed byRamakrishna.He had affinity for Hinduism. Andre Weil was an agnostic but respected religions. He often teased me aboutreincarnationin which he did not believe. He told me he would like to be reincarnated as a cat. He would often impress me by readings inBuddhism.
  35. ^"289085 Andreweil (2004 TC244)".Minor Planet Center.Retrieved11 September2019.
  36. ^"MPC/MPO/MPS Archive".Minor Planet Center.Retrieved11 September2019.
  37. ^Ore, Oystein (1936)."Book Review: Arithmétique et Géométrie sur les Variétés Algébriques".Bulletin of the American Mathematical Society.42(9): 618–619.doi:10.1090/S0002-9904-1936-06368-8.
  38. ^Cairns, Stewart S. (1939)."Review:Sur les Espaces à Structure Uniforme et sur la Topologie Générale,by A. Weil "(PDF).Bull. Amer. Math. Soc.45(1): 59–60.doi:10.1090/s0002-9904-1939-06919-X.Archived(PDF)from the original on 9 October 2022.
  39. ^Zariski, Oscar(1948)."Review:Foundations of Algebraic Geometry,by A. Weil "(PDF).Bull. Amer. Math. Soc.54(7): 671–675.doi:10.1090/s0002-9904-1948-09040-1.Archived(PDF)from the original on 9 October 2022.
  40. ^Chern, Shiing-shen(1950)."Review:Variétés abéliennes et courbes algébriques,by A. Weil ".Bull. Amer. Math. Soc.56(2): 202–204.doi:10.1090/s0002-9904-1950-09391-4.
  41. ^Weil, André (1974).Basic Number Theory.doi:10.1007/978-3-642-61945-8.ISBN978-3-540-58655-5.
  42. ^Weil, André (1971),Dirichlet Series and Automorphic Forms: Lezioni Fermiane,Lecture Notes in Mathematics, vol. 189,doi:10.1007/bfb0061201,ISBN978-3-540-05382-8,ISSN0075-8434
  43. ^Weil, André (1976).Elliptic Functions according to Eisenstein and Kronecker.doi:10.1007/978-3-642-66209-6.ISBN978-3-540-65036-2.
  44. ^Weil, André (1979).Number Theory for Beginners.New York, NY: Springer New York.doi:10.1007/978-1-4612-9957-8.ISBN978-0-387-90381-1.
  45. ^Humphreys, James E.(1983). "Review ofAdeles and Algebraic Groupsby A. Weil ".Linear & Multilinear Algebra.14(1): 111–112.doi:10.1080/03081088308817546.
  46. ^Ribenboim, Paulo(1985)."Review ofNumber Theory: An Approach Through History From Hammurapi to Legendre,by André Weil "(PDF).Bull. Amer. Math. Soc. (N.S.).13(2): 173–182.doi:10.1090/s0273-0979-1985-15411-4.Archived(PDF)from the original on 9 October 2022.
  47. ^Berg, Michael (1 January 2015)."Review ofŒuvres Scientifiques - Collected Papers,Volume 1 (1926–1951) ".MAA Reviews, Mathematical Association of America.
  48. ^Audin, Michèle(2011)."Review:At Home with André and Simone Weil,by Sylvie Weil "(PDF).Notices of the AMS.58(5): 697–698.Archived(PDF)from the original on 9 October 2022.

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