744:
332:. 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.
113:. 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.
359:
In 2018, when accepting the Abel Prize, Langlands delivered a paper reformulating the geometric program using tools similar to his original
Langlands correspondence.
303:
155:
259:
223:
748:
323:
175:
549:
328:
A claimed proof of the categorical unramified geometric
Langlands conjecture was announced on May 6, 2024 by a team of mathematicians including
478:(1983). "Two-dimensional ℓ–adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2)".
524:
44:
671:
394:
Lafforgue, Laurent (2002). "Chtoucas de
Drinfeld, formule des traces d'Arthur–Selberg et correspondance de Langlands".
480:
87:
86:
in the late 1960s, the
Langlands correspondence is related to important conjectures in number theory such as the
783:
754:
82:
is a collection of results and conjectures relating number theory and representation theory. Formulated by
773:
689:
Kapustin, Anton; Witten, Edward (2007). "Electric-magnetic duality and the geometric
Langlands program".
582:
778:
444:
608:
91:
79:
32:
353:
67:
270:
122:
110:
28:
708:
651:
232:
196:
117:
59:
8:
712:
655:
724:
698:
677:
641:
395:
308:
160:
48:
24:
681:
667:
475:
264:
178:
63:
728:
716:
659:
329:
83:
439:
182:
663:
499:
58:
The existence of the geometric
Langlands correspondence in the specific case of
720:
633:
442:(1987). "Correspondance de Langlands géométrique pour les corps de fonctions".
341:
226:
767:
345:
40:
36:
106:
98:
348:
described a connection between the geometric
Langlands correspondence and
102:
636:(2007). "Lectures on the Langlands Program and Conformal Field Theory".
550:"Incredible maths proof is so complex that almost no one can explain it"
703:
646:
400:
349:
743:
55:
asserts the existence of the geometric
Langlands correspondence.
609:"Об аналитическом виде геометрической теории автоморфных форм1"
758:
525:"Monumental Proof Settles Geometric Langlands Conjecture"
456:
408:
374:
372:
311:
273:
235:
199:
163:
125:
638:
Frontiers in Number Theory, Physics, and
Geometry II
420:
369:
583:"The Greatest Mathematician You've Never Heard Of"
317:
297:
253:
217:
193:The geometric Langlands conjecture was proved for
169:
149:
765:
97:Langlands correspondences can be formulated for
688:
267:proved the geometric Langlands conjecture for
66:in 2002, where it follows as a consequence of
500:"Proof of the geometric Langlands conjecture"
691:Communications in Number Theory and Physics
755:Quantum geometric Langlands correspondence
702:
645:
606:
522:
399:
393:
474:
335:
632:
547:
462:
426:
414:
378:
116:The geometric Langlands conjecture for
766:
438:
389:
387:
62:over function fields was proven by
13:
749:Geometric Langlands correspondence
21:geometric Langlands correspondence
14:
795:
736:
492:
384:
742:
600:
575:
566:
541:
523:Klarreich, Erica (2024-07-19).
481:American Journal of Mathematics
31:. It is a reformulation of the
640:. Springer. pp. 387–533.
548:Wilkins, Alex (May 20, 2024).
516:
468:
432:
292:
280:
248:
242:
212:
206:
144:
132:
78:In mathematics, the classical
53:geometric Langlands conjecture
1:
626:
616:Institute of Advanced Studies
105:), which are classified into
73:
47:and applying techniques from
664:10.1007/978-3-540-30308-4_11
7:
88:Taniyama–Shimura conjecture
10:
800:
721:10.4310/cntp.2007.v1.n1.a1
607:Langlands, Robert (2018).
39:appearing in the original
35:obtained by replacing the
445:Duke Mathematical Journal
188:
572:Kapustin and Witten 2007
362:
352:, a property of certain
80:Langlands correspondence
33:Langlands correspondence
504:people.mpim-bonn.mpg.de
298:{\displaystyle GL(n,K)}
150:{\displaystyle GL(n,K)}
747:Quotations related to
354:quantum field theories
340:In a paper from 2007,
319:
305:over a function field
299:
255:
219:
171:
157:over a function field
151:
111:global function fields
784:Representation theory
476:Drinfeld, Vladimir G.
336:Connection to physics
320:
300:
261:by Drinfeld in 1983.
256:
254:{\displaystyle GL(2)}
220:
218:{\displaystyle GL(1)}
172:
152:
118:general linear groups
92:Fermat's Last Theorem
60:general linear groups
29:representation theory
309:
271:
233:
197:
161:
123:
19:In mathematics, the
713:2007CNTP....1....1K
656:2005hep.th...12172F
94:as a special case.
68:Lafforgue's theorem
16:Mathematical theory
774:Algebraic geometry
315:
295:
251:
215:
177:was formulated by
167:
147:
49:algebraic geometry
25:algebraic geometry
779:Langlands program
673:978-3-540-30307-7
318:{\displaystyle K}
265:Laurent Lafforgue
179:Vladimir Drinfeld
170:{\displaystyle K}
90:, which includes
64:Laurent Lafforgue
791:
746:
732:
706:
685:
649:
620:
619:
613:
604:
598:
597:
595:
594:
579:
573:
570:
564:
563:
561:
560:
545:
539:
538:
536:
535:
520:
514:
513:
511:
510:
496:
490:
489:
472:
466:
465:, p. 31,46.
460:
454:
453:
436:
430:
424:
418:
412:
406:
405:
403:
391:
382:
376:
330:Dennis Gaitsgory
324:
322:
321:
316:
304:
302:
301:
296:
260:
258:
257:
252:
224:
222:
221:
216:
176:
174:
173:
168:
156:
154:
153:
148:
84:Robert Langlands
41:number theoretic
799:
798:
794:
793:
792:
790:
789:
788:
764:
763:
739:
674:
634:Frenkel, Edward
629:
624:
623:
611:
605:
601:
592:
590:
581:
580:
576:
571:
567:
558:
556:
546:
542:
533:
531:
529:Quanta Magazine
521:
517:
508:
506:
498:
497:
493:
473:
469:
461:
457:
437:
433:
425:
421:
417:, p. 3,24.
413:
409:
392:
385:
377:
370:
365:
338:
310:
307:
306:
272:
269:
268:
234:
231:
230:
198:
195:
194:
191:
162:
159:
158:
124:
121:
120:
76:
45:function fields
17:
12:
11:
5:
797:
787:
786:
781:
776:
762:
761:
752:
738:
737:External links
735:
734:
733:
704:hep-th/0604151
686:
672:
647:hep-th/0512172
628:
625:
622:
621:
599:
574:
565:
540:
515:
491:
467:
455:
440:Laumon, Gérard
431:
419:
407:
383:
367:
366:
364:
361:
342:Anton Kapustin
337:
334:
314:
294:
291:
288:
285:
282:
279:
276:
250:
247:
244:
241:
238:
227:Pierre Deligne
214:
211:
208:
205:
202:
190:
187:
166:
146:
143:
140:
137:
134:
131:
128:
75:
72:
15:
9:
6:
4:
3:
2:
796:
785:
782:
780:
777:
775:
772:
771:
769:
760:
756:
753:
750:
745:
741:
740:
730:
726:
722:
718:
714:
710:
705:
700:
696:
692:
687:
683:
679:
675:
669:
665:
661:
657:
653:
648:
643:
639:
635:
631:
630:
617:
610:
603:
588:
584:
578:
569:
555:
554:New Scientist
551:
544:
530:
526:
519:
505:
501:
495:
487:
483:
482:
477:
471:
464:
459:
451:
447:
446:
441:
435:
429:, p. 46.
428:
423:
416:
411:
402:
397:
390:
388:
380:
375:
373:
368:
360:
357:
355:
351:
347:
346:Edward Witten
343:
333:
331:
326:
312:
289:
286:
283:
277:
274:
266:
262:
245:
239:
236:
228:
209:
203:
200:
186:
184:
183:Gérard Laumon
180:
164:
141:
138:
135:
129:
126:
119:
114:
112:
108:
107:number fields
104:
100:
99:global fields
95:
93:
89:
85:
81:
71:
69:
65:
61:
56:
54:
50:
46:
42:
38:
37:number fields
34:
30:
26:
22:
751:at Wikiquote
697:(1): 1–236.
694:
690:
637:
615:
602:
591:. Retrieved
589:. 2018-11-15
586:
577:
568:
557:. Retrieved
553:
543:
532:. Retrieved
528:
518:
507:. Retrieved
503:
494:
485:
479:
470:
463:Frenkel 2007
458:
449:
443:
434:
427:Frenkel 2007
422:
415:Frenkel 2007
410:
401:math/0212399
381:, p. 3.
379:Frenkel 2007
358:
339:
327:
263:
192:
115:
103:local fields
101:(as well as
96:
77:
57:
52:
20:
18:
43:version by
768:Categories
627:References
593:2020-02-17
587:The Walrus
559:2024-07-09
534:2024-07-20
509:2024-07-09
452:: 309–359.
74:Background
682:119611071
488:: 85–114.
350:S-duality
325:in 2002.
185:in 1987.
729:30505126
229:and for
23:relates
709:Bibcode
652:Bibcode
51:. The
727:
680:
670:
189:Status
725:S2CID
699:arXiv
678:S2CID
642:arXiv
612:(PDF)
396:arXiv
363:Notes
759:nLab
668:ISBN
344:and
181:and
27:and
757:at
717:doi
660:doi
486:105
225:by
109:or
770::
723:.
715:.
707:.
693:.
676:.
666:.
658:.
650:.
614:.
585:.
552:.
527:.
502:.
484:.
450:54
448:.
386:^
371:^
356:.
70:.
731:.
719::
711::
701::
695:1
684:.
662::
654::
644::
618:.
596:.
562:.
537:.
512:.
404:.
398::
313:K
293:)
290:K
287:,
284:n
281:(
278:L
275:G
249:)
246:2
243:(
240:L
237:G
213:)
210:1
207:(
204:L
201:G
165:K
145:)
142:K
139:,
136:n
133:(
130:L
127:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.