Content deleted Content added
also known as the geometric Langlands conjecture |
Fixed typo Tags: Mobile edit Mobile web edit |
||
| (10 intermediate revisions by 3 users not shown) | |||
Line 1:
{{Short description|Mathematical theory}}
In mathematics, the '''geometric Langlands correspondence'''
The
==
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.
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.
==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
Line 15 ⟶ 20:
| date = 2002
| eprint = math/0212399
}}</ref>
}}</ref> A claimed proof of the geometric Langlands conjecture was announced on May 6th, 2024 by a team of mathematicians including [[Dennis Gaitsgory]].<ref name=":1" /> The proof is detailed by 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 [[Vladimir 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==
| |||