Mathematics

An introduction to the theory of derived functors in homological algebra, and its applications to sheaves and other geometric and algebraic objects.

• Basic notions: categories, functors, abelian categories, complexes
• Derived functors: injective objects, right derived functors, long exact sequence of a derived functor, acyclic resolutions, delta-functors.
• Introduction to sheaves: presheaves, sheaves, étalé space, direct and inverse images
• Cohomology of sheaves: Cech cohomology, sheaf cohomology, comparing Cech and sheaf cohomology
• Spectral sequences: filtered complexes, spectral sequences of a double complex, hypercohomology and hyperderived functors, applications (local-to-global, Cech, Leray, Künneth)

Prerequisites:  basic topology and abstract algebra

 

• U. Bruzzo, B. Graña Otero, Derived functors and sheaf cohomology, World Scientific
• R. B. Tennison, Sheaf Theory, Cambridge University Press
• R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris

The purpose of the course is to offer an introduction to advanced techniques in homological algebra that find applications in several areas, such as algebraic topology, algebraic and differential geometry, microlocal analysis, and more recently in more applied areas, for instance, machine learning to give one example.

The course is aimed at second year “laurea magistrate” mathematics students. While only basic topology and abstract algebra are strictly needed to attend the course, some knowledge in one or more fields such as algebraic topology, differential geometry, algebraic geometry will help the students to build intuition and construct examples.

The course will mainly consist of traditional lectures, during which the lecturer will help the student to participate in an interactive way. The lectures will include examples and problem solving session, with the active participation of the students. The exam will consist in the presentation of an advanced topic agreed with the lecturer in a talk approximately 50 minutes long.