Syllabus

Office Hours: 1:15-2:15 Monday, 2-3 Thursday, and by appointment.

The current plan is:

(3 weeks) categorical catch-up: additive, abelian, and triangulated categories, and functors between them; t-structures; derived categories.

(4 weeks) sheaf theory: sheaves and functors, locally constant and constructible sheaves, Verdier duality, gluing, the perverse t-structure.

(4 weeks) perverse sheaves on flag varieties: the flag variety, Bruhat decomposition, highest weight categories, projective cover of the skyscraper sheaf and its properties, convolution.

(3 weeks) Mixed sheaves and purity; graded versions of categories. Koszul Duality.

(1 week) Depending on interests of class; possibly seminar style introductions to some other geometric representation theory topics, for instance (1) parity sheaves (2) geometric Satake equivalence (3) Springer correspondence. 

Lecture Diary:

  • 1/05 - Super brief Kazhdan--Lusztig introduction; Reminders on categories (monic, epi, zero object, kernels, cokernels, functors, natural transformations, equivalence vs isomorphism of categories).
  • 1/08 - Additive and abelian categories, additive and exact functors, Yoneda's Lemma/Embedding, Freyd-Mitchell Embedding Theorem with proof sketch.
  • 1/10 - Triangulated categories, some Lemmas, cohomological functors
  • 1/12 - Chain complexes on an additive category, homotopy category, derived category
  • 1/15 - holiday
  • 1/17 - no class; UGA closed due to weather
  • 1/19 - distinguished triangles in homotopy category
  • 1/22 - derived category and derived functors
  • 1/24 - t-structure definition and truncation functors
  • 1/26 - heart of t-structure is abelian; matching Ext in heart with Hom(--, [1]) in triangulated category.
  • 1/29 - no class; power outage in Boyd
  • 1/31 - presheaves, sheaves definitions
  • 2/02 - sheafification, monomorphisms of sheaves and epimorphisms of sheaves versus stalks/sections
  • 2/05 - injectives, global sections, global sections with compact support, pullback
  • 2/07 - pushforward, proper pushforward, cartesian squares
  • 2/09 - extension by zero and its right adjoint, open/closed exact sequence and distinguished triangle
  • 2/12 - local systems
  • 2/14 - monodromy functor for locally constant sheaves
  • 2/16 - covering maps; singular cohomology
  • 2/19 - computations on the affine line
  • 2/21 - right adjoint to f_!, smooth morphism, Poincar\'e Duality
  • 2/23 - Stratifications, examples from group actions, constructible sheaves
  • 2/26 - Constructible sheaves, Verdier duality
  • 2/28 - Verdier duality on a point or for complexes on smooth variety with locally constant cohomology
  • 3/02 - Verdier duality is a duality
  • 3/05 - definition of perverse t-structure and first properties
  • 3/07 - perverse t-structure is a t-structure and exactness properties for push/pull along open/closed inclusion
  • 3/09 - intermediate extension
  • 3/19 - Intermediate extension continued 
  • 3/21 - Intersection cohomology complexes; perverse sheaves are noetherian
  • 3/23 - simple perverse sheaves; perverse sheaves have finite length; exactness results for special morphisms
  • 3/26 - the flag variety and Bruhat decomposition; Example: G = SL2; standard/costandard objects on the flag variety
  • 3/28 - highest weight categories; Bruhat constructible perverse sheaves on flag varieties is highest weight.
  • 3/30 - finish proof of enough projectives in highest weight categories; Hecke algebra
  • 4/02 - Hecke algebra; equivariant sheaves, equivariant perverse sheaves, and equivariant derived categories
  • 4/04 - Convolution on the flag variety, Bott--Samelson varieties
  • 4/06 - Convolution continued, parity considerations
  • 4/09 -Categorification theorem for the Hecke algebra
  • 4/11 - Categorification theorem continued
  • 4/13 - Variants of categorification theorem (parity sheaves, p-canonical basis, Soergel bimodules)
  • 4/16 - Connection with representation theory of semisimple Lie algebras
  • 4/18 - Koszul duality; graded versions of categories
  • 4/20 - existence and uniqueness of tilting objects in highest weight catgories
  • 4/23 - affine flag variety, affine Grassmannian, and the extended affine Hecke algebra
  • 4/25 - geometric Satake equivalence

Problem Sets:

  • PS1 (3-5 exercises seems reasonable. If you're newer to categories, I suggest starting with 1-3. Those more comfortable, may skip directly to 4-7.) Due 3/09.
  • PS2: Exercises 1.8.2, 1.8.3, 2.1.3, 2.1.9, 2.2.1, 2.2.2, 2.3.2  from P. Achar's text (3-5 seems reasonable) Due 3/23.
  • PS3: 2.13.1, 2.13.2, 2.13.5, 2.13.6, 3.9.1, 3.13.1. We did the first two of these in class. It isn't necessary that you turn in written versions of these, but you should try to undersand those two first before moving on to the others. Again, 3-5 seems reasonable. Due 4/06.

 

...