In mathematics, the geometric Langlands correspondence relates algebraic geometry and representation theory.
It 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.
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.
[2] A claimed proof of the categorical unramified geometric Langlands conjecture was announced on May 6, 2024 by a team of mathematicians including Dennis Gaitsgory.
[8][9] 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".
[10] 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.