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{{Short description|Mathematical theory}}
#REDIRECT [[Langlands program]]▼
In mathematics, the '''geometric Langlands correspondence''' relates [[algebraic geometry]] and [[representation theory]]. It is a reformulation of the [[Langlands correspondence]] obtained by replacing the [[number fields]] appearing in the original [[number theory|number theoretic]] version by [[function field of an algebraic variety|function fields]] and applying techniques from [[algebraic geometry]].{{sfn|Frenkel|2007|page=3}} The '''geometric Langlands conjecture''' asserts the existence of the geometric Langlands correspondence.
The existence of the geometric Langlands correspondence in the specific case of [[general linear group]]s over function fields was proven by [[Laurent Lafforgue]] in 2002, where it follows as a consequence of [[Lafforgue's theorem]].<ref name=":0" />
==Background==
In mathematics, the classical [[Langlands correspondence]] is a collection of results and conjectures relating number theory and representation theory. Formulated by [[Robert Langlands]] in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the [[Taniyama–Shimura conjecture]], which includes [[Fermat's Last Theorem]] as a special case.{{sfn|Frenkel|2007|page=3}}
Langlands correspondences can be formulated for [[global field]]s (as well as [[local field]]s), which are classified into [[number field]]s or [[global function field]]s. Establishing the classical Langlands correspondence, for number fields, has proven extremely difficult. As a result, some mathematicians posed the geometric Langlands correspondence for global function fields, which in some sense have proven easier to deal with.{{sfn|Frenkel|2007|page=3,24}}
The geometric Langlands conjecture for [[general linear group]]s <math>GL(n,K)</math> over a function field <math>K</math> was formulated by [[Vladimir Drinfeld]] and [[Gérard Laumon]] in 1987.{{sfn|Frenkel|2007|page=46}}<ref>{{cite journal |first=Gérard |last=Laumon |author-link=Gérard Laumon |title=Correspondance de Langlands géométrique pour les corps de fonctions |journal=[[Duke Mathematical Journal]] |volume=54 |year=1987 |pages=309-359}}</ref>
==Status==
The geometric Langlands conjecture was proved for <math>GL(1)</math> by [[Pierre Deligne]] and for <math>GL(2)</math> by Drinfeld in 1983.{{sfn|Frenkel|2007|page=31,46}}<ref>{{cite journal |first=Vladimir G. |last=Drinfeld |author-link=Vladimir Drinfeld |title=Two-dimensional ℓ–adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2) |journal=[[American Journal of Mathematics]] |volume=105 |year=1983 |pages=85–114}}</ref>
[[Laurent Lafforgue]] proved the geometric Langlands conjecture for <math>GL(n,K)</math> over a function field <math>K</math> in 2002.<ref name=":0">{{cite arXiv
| last = Lafforgue
| first = Laurent
| title = Chtoucas de Drinfeld, formule des traces d'Arthur–Selberg et correspondance de Langlands
| date = 2002
| eprint = math/0212399
}}</ref>
A claimed proof of the categorical unramified geometric Langlands conjecture was announced on May 6, 2024 by a team of mathematicians including [[Dennis Gaitsgory]].<ref name=":1">{{Cite web |title=Proof of the geometric Langlands conjecture |url=https://people.mpim-bonn.mpg.de/gaitsgde/GLC/ |access-date=2024-07-09 |website=people.mpim-bonn.mpg.de}}</ref><ref>{{Cite web |last=Klarreich |first=Erica |date=2024-07-19 |title=Monumental Proof Settles Geometric Langlands Conjecture |url=https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/ |access-date=2024-07-20 |website=Quanta Magazine |language=en}}</ref> The claimed proof is contained in more than 1,000 pages across five papers and has been called "so complex that almost no one can explain it". Even conveying the significance of the result to other mathematicians was described as "very hard, almost impossible" by Drinfeld.<ref>{{Cite web |last=Wilkins |first=Alex |date=May 20, 2024 |title=Incredible maths proof is so complex that almost no one can explain it |url=https://www.newscientist.com/article/2431964-incredible-maths-proof-is-so-complex-that-almost-no-one-can-explain-it/ |access-date=2024-07-09 |website=New Scientist |language=en-US}}</ref>
==Connection to physics==
In a paper from 2007, [[Anton Kapustin]] and [[Edward Witten]] described a connection between the geometric Langlands correspondence and [[S-duality]], a property of certain [[quantum field theories]].<ref>Kapustin and Witten 2007</ref>
In 2018, when accepting the Abel Prize, Langlands delivered a paper reformulating the geometric program using tools similar to his original Langlands correspondence.<ref>{{Cite web|url=https://thewalrus.ca/the-greatest-mathematician-youve-never-heard-of/|title=The Greatest Mathematician You've Never Heard Of|date=2018-11-15|website=The Walrus|language=en-US|access-date=2020-02-17}}</ref><ref>{{Cite web|url=https://publications.ias.edu/sites/default/files/iztvestiya_3.pdf|title=Об аналитическом виде геометрической теории автоморфных форм1|last=Langlands|first=Robert|date=2018|website=Institute of Advanced Studies|archive-url=|archive-date=|access-date=}}</ref> Langlands' ideas were further developed by Etingof, Frenkel, and Kazhdan.<ref>{{cite arXiv | author= Etingof, Pavel and Frenkel, Edward and Kazhdan, David | title = An analytic version of the Langlands correspondence for complex curves | date = 2019 | eprint = 1908.09677}}</ref>
==Notes==
{{reflist}}
==References==
* {{cite book |last1=Frenkel |first1=Edward |title=Frontiers in Number Theory, Physics, and Geometry II |chapter=Lectures on the Langlands Program and Conformal Field Theory |authorlink=Edward Frenkel |date=2007 |publisher=Springer |pages=387–533 |doi=10.1007/978-3-540-30308-4_11 |bibcode=2005hep.th...12172F |arxiv=hep-th/0512172 |isbn=978-3-540-30307-7 |s2cid=119611071 }}
* {{cite journal |last1=Kapustin |first1=Anton |last2=Witten |first2=Edward |title=Electric-magnetic duality and the geometric Langlands program |journal=Communications in Number Theory and Physics |volume=1 |issue=1 |pages=1–236 |date=2007 |doi=10.4310/cntp.2007.v1.n1.a1|arxiv = hep-th/0604151 |bibcode = 2007CNTP....1....1K |s2cid=30505126 }}
==External links==
*{{wikiquote-inline}}
*[https://ncatlab.org/nlab/show/quantum+geometric+Langlands+correspondence Quantum geometric Langlands correspondence] at [[nLab]]
[[Category:Algebraic geometry]]
[[Category:Representation theory]]
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