Applications of triangulated categories in representation theory.

Lead Research Organisation: University of Bristol
Department Name: Mathematics


Over the last few decades, triangulated categories have moved from being considered rather technical tools in algebraic geometry to assume central prominence as an object of study in many other algebraic areas of mathematics: notably representation theory of various kinds, commutative algebra and homotopy theory. In all of these fields, the language of triangulated categories has immensely clarified the ideas behind existing central problems, as well as raising important new problems.

The initial aim of the project will be to study some recent major theoretical advances and insights involving the triangulated categories that arise in the representation theory of finite dimensional algebras (derived categories and stable module categories), particularly the work of Rouquier on dimensions of triangulated categories, and very recent connections discovered between the unbounded derived category and the long-standing "homological conjectures". These developments have opened up a whole range of questions about generation of triangulated categories, many of which are open even in apparently simple cases. By initially trying to answer questions about these relatively simple examples it is expected that insight will be gained that can be applied to far more complicated and less well-understood examples.


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509619/1 01/10/2016 30/09/2021
2013880 Studentship EP/N509619/1 01/10/2017 30/09/2021 Charlotte Cummings Cummings
Description In representation theory there are a collection of conjectures called the homological conjectures which have been around since the 1980s and concern properties of the structure of algebraic mathematical objects. The work achieved proves that large collections of algebraic objects satisfy these conjectures by exploiting their relationship with simpler algebraic objects.
Exploitation Route In the academic world this work can be used to provide more examples of algebraic objects which satisfy the homological conjectures.
Sectors Other