Editing André Weil (section)

==Work==
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]]).<ref>A. Weil, ''L'arithmétique sur les courbes algébriques'', Acta Math 52, (1929) p.&nbsp;281–315, reprinted in vol 1 of his collected papers {{isbn|0-387-90330-5}}
.</ref> [[Mordell's theorem]] had an ''ad hoc'' proof;<ref>{{cite journal
| last1=Mordell |first1=L. J. |authorlink1=Louis Mordell
| title=On the rational solutions of the indeterminate equations of the third and fourth degrees
|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]
| volume=21
| year=1922
| pages=179–192
| url=https://archive.org/details/proceedingscambr21camb/page/178/mode/2up}}</ref> Weil began the separation of the [[infinite descent]] argument into two types of structural approach, by means of [[height function]]s 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 [[Weil conjectures|Riemann hypothesis for zeta-functions]] of curves over finite fields,<ref>{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Numbers of solutions of equations in finite fields | doi=10.1090/S0002-9904-1949-09219-4 | mr=0029393 | year=1949 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=55 | pages=497–508 | issue=5| doi-access=free }} Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil {{isbn|0-387-90330-5}}</ref> 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]],<ref>{{Citation | last1=Dwork | first1=Bernard | author1-link=Bernard Dwork | title=On the rationality of the zeta function of an algebraic variety | jstor=2372974 | mr=0140494 | year=1960 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=82 | pages=631–648 | doi=10.2307/2372974 | issue=3 | publisher=American Journal of Mathematics, Vol. 82, No. 3}}</ref> [[Alexander Grothendieck]],<ref>{{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Proc. Internat. Congress Math. (Edinburgh, 1958) | publisher=[[Cambridge University Press]] | mr=0130879 | year=1960 | chapter=The cohomology theory of abstract algebraic varieties | pages=103–118|chapter-url=http://grothendieckcircle.org/}}
</ref><ref>{{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Séminaire Bourbaki | chapter-url=http://www.numdam.org/item?id=SB_1964-1966__9__41_0 | publisher=[[Société Mathématique de France]] | location=Paris | mr=1608788 | year=1995 | volume=9 | chapter=Formule de Lefschetz et rationalité des fonctions L | pages=41–55|orig-year=1965}}
</ref><ref>{{citation
 | last = Grothendieck | first = Alexander | author-link = Alexander Grothendieck
 | doi = 10.1007/BFb0068688
 | isbn = 978-3-540-05987-5
 | mr = 0354656
 | publisher = Springer-Verlag
 | series = Lecture Notes in Mathematics
 | title = Groupes de monodromie en géométrie algébrique, I: Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I)
 | volume = 288
 | year = 1972}}</ref> [[Michael Artin]], and finally by [[Pierre Deligne]], who completed the most difficult step in 1973.<ref>{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Séminaire Bourbaki vol. 1968/69 Exposés 347–363 | chapter-url=http://www.numdam.org/item?id=SB_1968-1969__11__139_0 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-05356-9 | doi=10.1007/BFb0058801 | year=1971 | volume=179 | chapter=Formes modulaires et représentations l-adiques }}
</ref><ref>{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=La conjecture de Weil. I | url=http://www.numdam.org/item?id=PMIHES_1974__43__273_0 | mr=0340258 | year=1974 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=43 | issn=1618-1913 | issue=43 | pages=273–307| doi=10.1007/BF02684373 | s2cid=123139343 }}
</ref><ref>{{citation
 |editor-last=Deligne 
 |editor-first=Pierre 
 |editor-link=Pierre Deligne 
 |title=Cohomologie Etale 
 |series=Lecture Notes in Mathematics 
 |publisher=[[Springer-Verlag]] 
 |place=Berlin 
 |year=1977 
 |isbn=978-0-387-08066-6 
 |language=fr 
 |url=http://modular.fas.harvard.edu/sga/sga/4.5/index.html 
 |doi=10.1007/BFb0091516 
 |volume=569 
 |issue=569 
 |url-status=dead 
 |archive-url=https://web.archive.org/web/20090515034906/http://modular.fas.harvard.edu/sga/sga/4.5/index.html 
 |archive-date=15 May 2009 
}}
</ref><ref>{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=La conjecture de Weil. II | url=http://www.numdam.org/item?id=PMIHES_1980__52__137_0 | mr=601520 | year=1980 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=52 | issn=1618-1913 | issue=52 | pages=137–252| doi=10.1007/BF02684780 | s2cid=189769469 }}
</ref><ref>{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Katz | first2=Nicholas | author2-link=Nicholas Katz | title=Groupes de monodromie en géométrie algébrique. II | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics, Vol. 340 | isbn=978-3-540-06433-6 | doi=10.1007/BFb0060505 | mr=0354657 | year=1973 | volume=340}}</ref>

Weil introduced the [[adele ring]]<ref>A. Weil, ''Adeles and algebraic groups'', Birkhauser, Boston, 1982</ref> in the late 1930s, following [[Claude Chevalley]]'s lead with the [[idele]]s, and gave a proof of the [[Riemann–Roch theorem]] with them (a version appeared in his ''[[Basic Number Theory]]'' in 1967).<ref>{{citation|mr=0234930 |last=Weil|first= André |title=Basic number theory.|series=Die Grundlehren der mathematischen Wissenschaften |volume=144 |publisher=Springer-Verlag New York, Inc., New York |date=1967 |isbn= 3-540-58655-5 }}</ref> His 'matrix divisor' ([[vector bundle]] ''avant la lettre'') Riemann–Roch theorem from 1938 was a very early anticipation of later ideas such as [[moduli space]]s of bundles. The [[Weil conjecture on Tamagawa numbers]]<ref>{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Exp. No. 186, Adèles et groupes algébriques | url=http://www.numdam.org/item?id=SB_1958-1960__5__249_0 | series=Séminaire Bourbaki | year=1959 | volume=5 | pages=249–257}}</ref> 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<ref name="ShimuraTaniyama">{{cite web | title=Some History of the Shimura-Taniyama Conjecture | first=Serge | last=Lang | pages=1301–1307 | url=https://www.ams.org/journals/notices/199511/199511FullIssue.pdf | access-date=2025-03-30}}</ref> from [[Serge Lang]] (resp. of [[Jean-Pierre 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.<ref>Lang, S. [https://www.ams.org/notices/199511/forum.pdf "Some History of the Shimura-Taniyama Conjecture."] Not. Amer. Math. Soc. 42, 1301–1307, 1995</ref>

Other significant results were on [[Pontryagin duality]] and [[differential geometry]].<ref>{{cite journal|author=Borel, A.|title=André Weil and Algebraic Topology|journal=Notices of the AMS|year=1999|volume=46|issue=4|pages=422–427|url=https://www.ams.org/notices/199904/borel.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/notices/199904/borel.pdf |archive-date=2022-10-09 |url-status=live}}</ref> 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 also chose the symbol [[null sign|∅]], derived from the letter [[Ø]] in the [[Norwegian alphabet]] (which he alone among the Bourbaki group was familiar with), to represent the [[empty set]].<ref>{{Cite web|first=Jeff|last=Miller|date=1 September 2010 |url=http://jeff560.tripod.com/set.html|title=Earliest Uses of Symbols of Set Theory and Logic|publisher=Jeff Miller Web Pages|access-date=21 September 2011}}</ref>

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]] 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]] by [[Irving Segal]] and [[David Shale]], gave a contemporary framework for understanding the classical theory of [[quadratic form]]s.<ref>{{cite journal |first=A. |last=Weil |title=Sur certains groupes d'opérateurs unitaires |language=fr|journal=Acta Math. |volume=111 |year=1964 |pages=143–211 |doi=10.1007/BF02391012|doi-access=free }}</ref> This was also a beginning of a substantial development by others, connecting [[representation theory]] and [[theta function]]s.

Weil was a member of both the [[National Academy of Sciences]]<ref>{{Cite web|title=Andre Weil|url=http://www.nasonline.org/member-directory/deceased-members/45882.html|access-date=2021-12-20|website=www.nasonline.org}}</ref> and the [[American Philosophical Society]].<ref>{{Cite web|title=APS Member History|url=https://search.amphilsoc.org/memhist/search?creator=Andr%C3%A9+Weil&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced|access-date=2021-12-20|website=search.amphilsoc.org}}</ref>