Introductory description

This is a topics module. Such modules are designed to address material of particular interest in the year of delivery. By their nature, the specific topics rotate, which means that the course will also be of use to PhD students in later years, even if they took the course for credit already. However, in each year the topics will be in algebra.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Possible topics comprise:

-Some basic notions from category theory, including additive and abelian categories and functors in abelian categories
-Simplicial sets
-Localisation and derived category of an abelian category
-Derived functors
-Spectral sequences and derived functor of a composition
-Triangulated categories and exact functors
-Exceptional sequences and semiorthogonal decompositions
-Cores and t-structures
-Some view towards dg- categories and A-infinity categories

We will also make every attempt to bring this material to life with lots of examples and applications in other fields of mathematics. Which areas these will mainly be taken from (algebraic geometry/coherent sheaves on varieties, representation theory, topology), will depend on the background of the participants.

Learning outcomes

By the end of the module, students should be able to:

• command fundamental tools from the theory of derived and triangulated categories
• understand of a selection of the concepts, theorems and calculations related to triangulated and derived categories
• apply properties and features of triangulated and derived categories to problems in other fields of mathematics.

Indicative reading list

-S.I. Gelfand, Y.I. Manin: Methods of Homological Algebra, 2nd ed., Springer Monographs in Math. (2003)
-D. Huybrechts: Fourier-Mukai Transforms in Algebraic Geometry, Oxford Math. Monographs (2006)
-A. Yekutieli, Derived categories, Cambridge studies in advanced math. 183, (2020)
-P. Seidel, Fukaya categories and Picard-Lefschetz theory, European Math. Society (2008)

Subject specific skills

The students will acquire the basic skills needed to apply derived category techniques in various areas of mathematics such as algebraic and complex analytic geometry, representation theory, and algebraic topology. They will learn to view and approach problems in those areas from the point of view of advanced homological algebra and triangulated categories. In various instances, this will also provide them with a unifying perspective on seemingly unrelated questions in different parts of mathematics.

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organising information
• presentation skills (both oral and written)