The geometric Langlands conjecture
Bundles, Hecke correspondences, geometric class field theory and the rank-one categorical correspondence, together with all-rank averaging, opers, critical-level calculations, spectral categories, categorical variants, construction of the Langlands functor, critical Kac-Moody localization, Eisenstein reduction, ambidexterity, generic opers, multiplicity one and the full conditional proof assembly, leading to the proof of the unramified conjecture.
15 draft lessons are available from the fifteen-lesson course. Mathematical hypotheses, proved conclusions and required inputs are stated in each lesson.
- From automorphic functions to automorphic sheaves (editable source)
- The moduli stack of bundles (editable source)
- Sheaves and D-modules on Bun_G (editable source)
- Hecke functors and Hecke eigensheaves (editable source)
- Geometric class field theory (editable source)
- The GL_1 case as an equivalence of categories (editable source)
- The unramified correspondence for GL_n: Drinfeld, Laumon, Frenkel–Gaitsgory–Vilonen (editable source)
- Opers, critical level and the Beilinson–Drinfeld construction (editable source)
- The spectral side: local systems, singular support and IndCoh_Nilp (editable source)
- The categorical conjecture and its variants (editable source)
- Constructing the Langlands functor (GLC I) (editable source)
- Kac-Moody localization and the fundamental local equivalence (GLC II) (editable source)
- Eisenstein series and the reduction to the cuspidal part (GLC III) (editable source)
- Ambidexterity, opers and multiplicity one (GLC IV and V) (editable source)
- The proof as a map (editable source)