This is my Master's dissertation, which was supervised by Dr. Pavel Safronov. In it, we covered foundational topics in representation theory and quantum algebra, such as Hopf algebras and their representations, monoidal categories, and the simplest example of a quantum group, Uq(sl2). These topics were motivated by applications of quantum groups and ribbon categories to the construction of link invariants.

This is my honours project, which was completed as a group project with three other fourth year students, and was supervised by Prof. Agata Smoktunowicz. We focused primarily on the work of Wolfgang Rump, who invented both cycle sets and braces as objects for studying set-theoretic solutions to the Yang-Baxter equation. Detailed proofs were given of Rump's results, particularly on cycle sets, which were initially only outlined by Rump, or described in not much detail. I contributed to all chapters, but particularly chapters 3 and 4, with chapter 4 being entirely my work.

This was the final assessment for a masters course in complex geometry, delivered by Prof. Brian R. Williams. It was completed as a group project with one other student, and was on Chern classes, which are invariants of complex vector bundles. We covered the fully general definition of Chern classes given in terms of invariant functions and curvature forms, and also examined how the Chern class of a holomorphic line bundle can be defined in terms of the sheaf cohomology. Although both of us contributed to researching all parts of the project, sections 1 and 4 were written entirely by myself, while 2 and 3 were written by my partner.

This was the final assessment for a reading course in homological algebra, organised by Dr. Pavel Safronov. The course was based on Charles A. Weibel's "An introduction to homological algebra", and for the final project I was tasked with writing a report based on a chapter we had not covered. I chose chapter 7, which was on the Lie algebra homology and cohomology, and I covered the definition of the homology/cohomology in terms of the invariant/coinvariant functors, the Chevalley-Eilenberg complex, and the full classification of the homology and cohomology for semisimple Lie algebras.

This was the final assessment for a reading course in Lie algebras, organised by Prof. Iain Gordon. I chose to look at the sl2 spider, which is a diagramatic way of representing invariant functions on sl2 representations. I gave some background on invariant functions, as well as the category of representations of a Lie algebra, and explained how invariant functions relate to hom-spaces. I then examined the construction of the sl2 spider, giving my own suggestions on how to make certain aspects of the construction less ambiguous, and noting connections to the Temperley-Lieb algebra.