Day 1:
Lecture 1.1: Road Map
Contents:
a) Approximate statement of the Geometric Langlands conjecture
b) Whittaker vs Evaluation compatibility
c) Kac-Moody vs Oper compatibility
d) The Fundamental diagram
Prerequisites: D-modules (assumed) , Algebraic stacks (assumed), the stack Bun(G) (assumed)
Corresponds to Sects. 0 & 1. of the Outline.
Lecture 1.2: D-modules and Quasi-Coherent sheaves on prestacks
Contents:
a) What is a prestack?
b) Quasi-coherent sheaves on prestack
c) The de Rham space of a prestack
d) D-modules as crystals (=quasi-coherent sheaves on the de Rham pre stack)
Prerequisites: Infinity-categories (assumed), DG categories (assumed), D-modules (assumed).
Lecture 1.3: Singular Support-I
Singular support on schemes
Contents:
a) Singular support of coherent sheaves via cohomological operations
b) Shifted cotangent `bundle'
c) Ind-coherent sheaves and singular support
d) Categories given by singular support
Prerequisites: DAG (assumed), Ind-coherent sheaves (will be introduced)
Corresponds to Sect. 2 of the Outline.
Day 2:
Lecture 2.1: Singular Support-II
Singular support on stacks
Contents:
a) Natural functors on ind-coherent sheaves
b) Functoriality of singular support
c) Ind-coherent sheaves on stacks
d) Formulation of the categorical Langlands conjecture
Prerequisites: Lecture 1.3, Algebraic stacks (assumed), the stack Bun(G) (assumed)
Corresponds to Sects 7, 9, 10 & 12 of Singular Support.
Lecture 2.2: Factorization-I (factorization spaces and algebras).
Contents:
a) Affine Grassmannian
b) Factorization algebras (linearization, examples)
c) Commutative factorization algebras
d) Ran space
e) Factorization homology
f) Contractibility of Ran space
Prerequisites: Lecture 1.2, Affine Grassmannian (will be introduced), factorization algebras (will be introduced).
Lecture 2.3: Spaces of rational maps to the flag variety
Contents:
a) The usual stack of B-bundles and the adelic picture
b) The stack of G-bundles equipped with a generic reduction to B (and its adelic picture)
c) Relation to Drinfeld's compactification
d) Adding the datum of a rational character of the adelic N
e) Allowing the character to degenerate
Prerequisites: Lecture 1.2, Algebraic stacks (assumed), adele groups (will be briefly introduced)
Corresponds to Sects. 5.1, 5.2, 7.1.1, 8.1, 8.2.1 & 8.2.2 of the Outline.
Day 3:
Lecture 3.1: Factorization-II (factorization categories)
Contents:
a) Categories over prestacks
b) Notion of factorization category
c) D-modules on the Affine Grassmannian as a factorization category
d) Representations of an algebraic group as a factorization category
e) Factorization algebras in factorization categories
Prerequisites: Lecture 1.2 & 2.2, DG categories, D-modules.
Lecture 3.2: Hecke action
Contents:
a) G(O), G(K) as factorization group-schemes
b) The Hecke category and action on D-modules on Bun(G)
c) Naive Satake equivalence and compatbility with the Geometric Langlands Conjecture
d) The (derived) Hecke category on the spectral side
e) Statement of derived (non-naive) Satake
Prerequisites: Lectures 1.2 & 3.1, Loop groups (will be introduced), the stack Bun(G), Geometric Satake (will be introduced), DAG.
Corresponds to Sect. 4 of the Outline.
Day 4:
Lecture 4.1: Factorization-III (local-to-global principle)
Contents:
a) Embedding of QCoh(LocSys(G)) into the Ran version of Rep(G)
b) Description of D-modules on LocSys(G) via the derived spectral Hecke action
c) Embedding of D-mod(Bun(G)) into the Ran version of D-mod(Gr(G))
Prerequisites: Lectures 1.2 & 3.1, the stack Bun(G), DG categories, D-modules.
Lecture 4.2: Global Whittaker and principal series categories
Contents:
a) Adelic picture: definition of spaces of functions and operation of Fourier transform
b) Definition of global Whittaker and principal series categories
c) Functors of Whittaker coefficient and constant term
d) Allowing the character to degenerate (degenerate and extended Whittaker categories)
Lectures 1.2 & 2.3, the stack Bun(G), DG categories, D-modules.
Corresponds to Sects. 5.3-5.8, 6.1, 7.1 & 8.2 of the Outline.
Lecture 4.3: Kac-Moody representations
Contents:
a) Definition of the DG category of represntations of the KM algebra
b) The Kazhdan-Lusztig category
c) The variant with moving points
d) The "down-to earth" localization picture.
Prerequisites: Loop groups, BB localization (assumed), D-modules on algebraic stacks, affine BB localization (will be sketched).
Corresponds to Sect. 10.1 of the Outline.
Day 5:
Lecture 5.1: Langlands duality for local and global Whittaker categories
Contents:
a) Definition of the local Whittaker category
b) Local-to-global connection in the adelic picture
c) Fully faithful embedding of the global Whittaker category into the local one
d) Langlands dual description of the local category
Prerequisites: Lectures 3.1, 4.1 and 4.2, DG categories, D-modules, Geometric Satake
Corresponds to Sect. 5.10 of the Outline.
Lecture 5.2: Langlands duality for local and global principal series categories
Contents:
a) Definition of the local principal series category
b) Langlands dual description of the local principal series category
c) Statement of the spectral description of the global principal series category
d) Deduce c) from b).
e) Statement for the degenerate version
Prerequisites: Lectures 3.1, 4.1 & 4.2, DG categories, D-modules, Geometric Satake.
Corresponds to Sects. 6.5 & 7.4 of the Outline.
Lecture 5.3: Langlands duality for the extended Whittaker category (gluing)
Contents:
a) Gluing of (DG) categories
b) The extended Whittaker category via gluing (for GL(2))
c) Gluing on the spectral side of the Langlands correspondence (for GL(2))
d) Gluing conjectures for general reductive group
Prerequisites: Lectures 4.2, 5.1 & 5.2, D-modules, DG categories, Geometric Satake, Ind-Coherent sheaves.
Corresponds to Sect. 9 of the Outline.
Lecture 5.4: Localization of Kac-Moody representations
Contents:
a) Localization picture via the infinitesimal groupoid
b) Factorizations algebras with Kac-Moody symmetry and their localization
c) Localization of the chiral algebra of differential operators
Prerequisites: Lectures 2.2, 3.1, 4.1 & 4.3, D-modules, DG categories, Affine Grassmannian, factorization algebras.
Day 6:
Lecture 6.1: Opers
Contents:
a) Definition of oper on a curve/formal disc
b) Opers without monodromy (local and global)
c) Relation to Kac-Moody representations
Prerequisites: Lecture 2.3, the notion of oper (will be introduced)
Corresponds to Sects. 10.2-10.4 of the Outline.
Lecture 6.2: Fundamental diagram and the proof of the Geometric Langlands Conjecture
Contents:
a) Reminder: the geometric Langlands conjecture and gluing
b) Generators of the categories on the two sides of the geometric Langlands conjecture
c) Construction of the Langlands functor
d) Proving that the Langlands functor is an equivalence
Prerequisites: Lectures 5.3, 5.4 & 6.1.
Corresponds to Sect. 11 of the Outline.