Editing Langlands program (section)

===Geometric conjectures===
{{main|Geometric Langlands correspondence}}

The geometric Langlands program, suggested by [[Gérard Laumon]] following ideas of [[Vladimir Drinfeld]], arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relates {{mvar|l}}-adic representations of the [[étale fundamental group]] of an [[algebraic curve]] to objects of the [[derived category]] of {{mvar|l}}-adic sheaves on the [[moduli stack of bundles|moduli stack]] of [[vector bundle]]s over the curve.

A 9-person collaborative project led by [[Dennis Gaitsgory]] announced a proof of the (categorical, unramified) geometric Langlands conjecture leveraging [[Hecke eigensheaves]] as part of the proof.<ref>{{cite web
| url=https://people.mpim-bonn.mpg.de/gaitsgde/GLC/
| last1=Gaitsgory | first1=Dennis | authorlink1=Dennis Gaitsgory
| title=Proof of the geometric Langlands conjecture
| access-date=August 19, 2024}}</ref><ref>{{cite arXiv
| last1=Gaitsgory | first1=Dennis | authorlink1=Dennis Gaitsgory
| last2=Raskin | first2=Sam
| date=May 2024
| title=Proof of the geometric Langlands conjecture I: construction of the functor
| eprint=2405.03599
| class=math.AG}}</ref><ref>{{cite arXiv
| last1=Arinkin | first1=D.
| last2=Beraldo | first2=D.
| last3=Campbell | first3=J.
| last4=Chen | first4=L.
| last5=Faergeman | first5=J.
| last6=Gaitsgory | first6=D.
| last7=Lin | first7=K.
| last8=Raskin | first8=S.
| last9=Rozenblyum | first9=N.
| date=May 2024
| title=Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE
| eprint=2405.03648
| class=math.AG}}</ref><ref>{{cite web |url=https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/ |title= Monumental Proof Settles Geometric Langlands Conjecture|date=July 19, 2024 |publisher=Quanta Magazine}}</ref>