MATH 7400: Homological Algebra and Applications (Spring 2011)

Instructor: Yuri Berest

This course is an introduction to homological algebra with a view towards applications in topology and geometry. The first (introductory) part will cover (roughly) Chapters III and IV of Gelfand-Manin's Methods of Homological Algebra. In the second part, we will look at applications of homological ideas in modern geometry and singularity theory, following A. Dimca's book Sheaves in Topology (Springer-Verlag, 2004). Topics will include:

1. Derived and triangulated categories. Derived functors.
2. Derived categories in topology (generalities on sheaves, derived tensor products, direct and inverse images, adjunction maps, local systems).
3. Poincaré-Verdier Duality (cohomological dimension of rings and spaces, the functor f ^!, Poincaré and Alexander duality, vanishing theorems).
4. Vanishing cycles and characteristic varieties (constructible sheaves, nearby and vanishing cycles, characteristic varieties and cycles).
5. Perverse sheaves (t-structures and abelian cores, definition of perverse sheaves, recollement, Beilinson's Gluing Theorem, D-modules and perverse sheaves, intersection cohomology).