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Abelova cena v roce 2018 udělena za Langlandsův program

Vítězslav Kala (2018)

Pokroky matematiky, fyziky a astronomie

V článku motivujeme a vysvětlíme základy Langlandsova programu, sítě domněnek propojujících řadu různých oblastí matematiky. Během toho se také setkáme s Riemannovou hypotézou a domněnkou Birche a Swinnerton-Dyera, dvěma ze sedmi problémů tisíciletí vyhlášených Clayovým matematickým institutem.

Compatibility of the theta correspondence with the Whittaker functors

Vincent Lafforgue, Sergey Lysenko (2011)

Bulletin de la Société Mathématique de France

We prove that the global geometric theta-lifting functor for the dual pair ( H , G ) is compatible with the Whittaker functors, where ( H , G ) is one of the pairs ( S 𝕆 2 n , 𝕊 p 2 n ) , ( 𝕊 p 2 n , S 𝕆 2 n + 2 ) or ( 𝔾 L n , 𝔾 L n + 1 ) . That is, the composition of the theta-lifting functor from H to G with the Whittaker functor for G is isomorphic to the Whittaker functor for H .

Dualité de Langlands quantique

Vadim Schechtman (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

Un survol des conjectures de Drinfeld, Beilinson, Gaitsgory et al. et de résultats de Gaitsgory sur la correspondance de Langlands quantique.

Geometric theta-lifting for the dual pair 𝕊𝕆 2 m , 𝕊 p 2 n

Sergey Lysenko (2011)

Annales scientifiques de l'École Normale Supérieure

Let X be a smooth projective curve over an algebraically closed field of characteristic  > 2 . Consider the dual pair H = SO 2 m , G = Sp 2 n over X with H split. Write Bun G and Bun H for the stacks of G -torsors and H -torsors on X . The theta-kernel Aut G , H on Bun G × Bun H yields theta-lifting functors F G : D ( Bun H ) D ( Bun G ) and F H : D ( Bun G ) D ( Bun H ) between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non ramified case)....

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