Geometric Langlands correspondence
In mathematics, the geometric Langlands correspondence is a reformulation of the
number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic geometry.[1] The geometric Langlands correspondence relates algebraic geometry and representation theory
.
The specific case of the geometric Langlands correspondence for general linear groups over function fields was proven by Laurent Lafforgue in 2002, where it follows as a consequence of Lafforgue's theorem.
History
In mathematics, the classical
Taniyama–Shimura conjecture, which includes Fermat's Last Theorem as a special case.[1] Establishing the Langlands correspondence in the number theoretic context has proven extremely difficult. As a result, some mathematicians have posed the geometric Langlands correspondence.[1]
Langlands correspondences can be formulated for
global function fields
. The classical Langlands correspondence is formulated for number fields. The geometric Langlands correspondence is instead formulated for global function fields, which in some sense have proven easier to deal with.
In 2002, the geometric Langlands correspondence was proven for general linear groups over a function field by Laurent Lafforgue.[2]
Connection to physics
In a paper from 2007,
quantum field theories.[3]
In 2018, when accepting the Abel Prize, Langlands delivered a paper reformulating the geometric program using tools similar to his original Langlands correspondence.[4][5]
Notes
- ^ a b c Frenkel 2007, p. 3
- arXiv:math/0212399.
- ^ Kapustin and Witten 2007
- ^ "The Greatest Mathematician You've Never Heard Of". The Walrus. 2018-11-15. Retrieved 2020-02-17.
- ^ Langlands, Robert (2018). "Об аналитическом виде геометрической теории автоморфных форм1" (PDF). Institute of Advanced Studies.
References
- S2CID 119611071.
- Kapustin, Anton; Witten, Edward (2007). "Electric-magnetic duality and the geometric Langlands program". Communications in Number Theory and Physics. 1 (1): 1–236. S2CID 30505126.
External links
Quotations related to Geometric Langlands correspondence at Wikiquote- Quantum geometric Langlands correspondence at nLab
