homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
Examples
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
In QFT and String theory
The categorical geometric Langlands conjecture is a higher category theoretic version of the geometric Langlands conjecture. It is formulated as an equivalence of stable (infinity,1)-categories of
and
where is a smooth complete curve over a field of characteristic zero, and is a reductive group and is its Langlands dual.
Gurbir Dhillon, An informal introduction to categorical representation theory and the local geometric Langlands program (arXiv:2205.14578)
Dennis Gaitsgory, Nick Rozenblyum, A study in derived algebraic geometry.
Dennis Gaitsgory et. al., Geometric representation theory, graduate seminar, Fall 2009–Spring 2010, web.
A precise statement of the categorical version of geometric Langlands conjecture (2011) [MO:q56571]
Proof:
Dennis Gaitsgory: Proof of the geometric Langlands conjecture (2025) [webpage]
Dennis Gaitsgory, Sam Raskin: Proof of the geometric Langlands conjecture I: construction of the functor [arXiv:2405.03599, pdf]
Dima Arinkin, D. Beraldo, Justin Campbell, L. Chen, Dennis Gaitsgory, J. Faergeman, Kevin Lin, Sam Raskin, Nick Rozenblyum: Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE [arXiv:2405.03648, pdf]
Justin Campbell, Lin Chen, Joakim Faergeman, Dennis Gaitsgory, Kevin Lin, Sam Raskin, Nick Rozenblyum: Proof of the geometric Langlands conjecture III: compatibility with parabolic induction [arXiv:2409.07051, pdf]
Dima Arinkin, D. Beraldo, L. Chen, J. Faergeman, Dennis Gaitsgory, Kevin Lin, Sam Raskin, Nick Rozenblyum: Proof of the geometric Langlands conjecture IV: ambidexterity [arXiv:2409.08670, pdf]
Dennis Gaitsgory, Sam Raskin: Proof of the geometric Langlands conjecture V: the multiplicity one theorem [arXiv:2409.09856, pdf]
Review:
Last revised on April 29, 2025 at 18:08:36. See the history of this page for a list of all contributions to it.