8/28/25 -- Patrick

Abstract: Definitions of the category of (co)chain complexes of an abelian category, and examples. Homotopies, homotopy equivalences and quasi-isomorphisms of (co)chain complexes. Different notions of localization of categories, definitions of the homotopy category and derived category of an abelian category. Examples and discussion of representations of objects and morphisms in the derived category.

9/4/25 -- Alex -- (Notes)

Abstract:  Poincare duality, Borel-Moore homology, examples and pictures.

9/11/25 -- Thomas -- (Notes)

Abstract:  Definitions of exactness of sequences and functors, derived functors, exactness of complexes of sheaves, the global sections functor, sheaf cohomology (w/ constant coefficients), _______________ resolutions (acyclic, injective, flabby, soft).

9/18/25 -- Thomas/Sam -- (Notes)

Abstract:  Continuation from last week, examples of computation of sheaf cohomology using resolution by the DeRham complex. Notions of resolutions in context of left derived functors (projective, flat, free).

Examples from algebraic/complex geometry, derived tensor product of modules.

9/25/25 -- Ziyal -- (Notes)

Abstract: The direct image and inverse image functors on sheaves, local and global properties, exactness, adjuctions, examples.

10/2/25 -- Ziyal -- (Notes)

Abstract:  The proper direct image functor on categories of sheaves, properties and relationships with other functors on sheaves, examples, proper base change.

10/9/25 -- Sam -- (Exercises) (Notes)

Abstract:  Application of functors on categories of sheaves, examples, functoriality for cohomology, computation of local cohomology.

10/16/25 -- Sam -- (Notes)

10/23/25 -- Jonathan -- (Notes)

Abstract: Sheaves on locally simply connected spaces, correspondence between locally constant sheaves and functors out of the fundamental groupoid.

10/30/25 -- Jonathan -- (Notes)

Abstract: Continuation from last week, identifying certain categories of locally constant sheaves: covering spaces, vector bundles with flat connection (Riemann-Hilbert correspondence); example of computing sheaf cohomology of the circle using Čech complex.

11/6/25 --  Thomas -- (Notes) 

11/13/25 -- Alex -- (Notes)

Abstract:  Stratified spaces, exit paths; exodromy and the correspondence between constructible sheaves and functors out of the exit path category.

11/20/25 --  Jonathan -- (Notes)