Geometric Langlands correspondence

In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic geometry. The geometric Langlands correspondence relates algebraic geometry and representation theory. The geometric Langlands conjecture asserts the existence of the geometric Langlands correspondence.

The existence of the geometric Langlands correspondence in the specific case of general linear groups over function fields was proven by Laurent Lafforgue in 2002, where it follows as a consequence of Lafforgue's theorem. A claimed proof of the geometric Langlands conjecture was announced on May 6th, 2024.

History
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. Establishing the Langlands correspondence in the number theoretic context has proven extremely difficult. As a result, some mathematicians have posed the geometric Langlands correspondence.

Langlands correspondences can be formulated for global fields (as well as local fields), which are classified into number fields or 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.

Laurent Lafforgue proved the geometric Langlands conjecture for general linear groups $$GL(n,K)$$ over a function field $$K$$ in 2002. A claimed proof of the geometric Langlands conjecture was announced on May 6th, 2024 by a team of mathematicians including Dennis Gaitsgory. 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.

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.

In 2018, when accepting the Abel Prize, Langlands delivered a paper reformulating the geometric program using tools similar to his original Langlands correspondence.