Pure or applied algebra from representation theory, category theory, and topology (GRANTJU17SF)

Representation theory is the study of algebraic objects, such as groups or rings, by investigating how they act on simpler objects such as vector spaces. In some nice situations, such as when considering representations of finite groups on complex vector spaces, we can break each representation into a direct sum of simple representations, which have no subrepresentations. But in other situations there are representations which are not simple yet cannot be broken down any more via direct sums. This forces us to work with quotients and extensions of representations. We thus gain a deeper understanding of representation theory, and the algebraic techniques

The study of extensions is part of homological algebra, a powerful theory of obstructions which comes from homology and homotopy groups in algebraic topology. This theory can be used to study algebras, or even to construct new algebras from old. A strong understanding of category theory is often important here, both to formulate precise mathematical statements and because the graphical calculus can help with performing computation. Related theories, such as operads, can be used to give precise duality theories or connect different algebraic structures.

A PhD project under my supervision is likely to involve some aspects of homological or homotopical algebra, though the precise project would depend on the interests of the student. Pure projects in areas such as representation theory of finite-dimensional algebras or related category theory are certainly welcomed, but projects which apply algebraic or categorical techniques to other areas including quantum information could also be possible.

The project may be available at an earlier start date of 1 April or 1 July 2017 but should be discussed with the primary supervisor in the first instance.