Geometric Langlands Conjecture
The Geometric Langlands Conjecture is a geometric version of the Langlands Program. It was proposed in the 1980s and was precisely stated by Dennis Gaitsgory and Dima Arinkin at the beginning of this century. They proposed this statement in a paper of more than 150 pages. The core idea is to find an equivalence relation that connects the category of D-modules (solutions of differential equations on certain spaces) of G-bundles on algebraic curves X with the Ind-Coh category (including all Ind-cohomology objects) of the local system of the Langlands dual group. This statement lays the theoretical foundation for proving the Geometric Langlands Conjecture. The Langlands Program itself was proposed by Canadian mathematician Robert P. Langlands in 1967, who first proposed this concept in a letter to André Weil.
The proof of the geometric Langlands conjecture was completed in 2024 by a team of nine mathematicians, including Chinese scholar Chen Lin. The team was led by Harvard University professor Dennis Gaitsgory and Yale University professor Sam Raskin. The final proof consists of five papers and more than 800 pages.
The five papers are:
- GLC I: Construction of the functor
- GLC II: Kac-Moody localization and the FLE
- The geometric Langlands functor II: equivalence on the Eisenstein part (this paper is in the process of being split into two: GLC-II and GLC-III)
- GLC IV: Ambidexterity
- GLC V: The multiplicity one theorem