PhD Studentship: Categorification in Representation Theory

Primary Supervisor – Prof. Vanessa Miemietz  

In the last 20 years, categorification has started to play an important role in representation theory and low-dimensional topology. For example, categorifications of algebras such as quantum groups and Hecke algebras have led to proofs or counterexamples of major conjectures, such as proofs of Broué’s abelian defect group conjecture for symmetric groups and the Kazhdan—Lusztig conjectures for all Coxeter types and counterexamples to James’ conjecture and Lusztig’s conjecture. Similarly, they have produced categorifications of knot invariants, which are strictly stronger than their decategorified version.  

This success has led to a desire to understand the categorified structures by studying their representation theory. Where in classical representation theory an algebra acts on vector spaces by linear transformations, in the categorified version a 2-category (categorifying the algebra) acts on categories (categorifying the vector space) by functors. Since functors have morphisms between them (so-called natural transformations), we obtain an extra layer of data that gives additional information about the algebra we originally wanted to study.   

This PhD project will investigate questions in the abstract 2-representation theory of the kind of 2-categories that appear in categorification and apply the results to specific examples. The work will thus take place at the interface of representation theory and category theory. 

Funding Details

Additional Funding Information

This PhD project is in a competition for a Faculty of Science funded studentship. Funding is available to UK applicants and comprises ‘home’ tuition fees and an annual stipend of £19,237 (for a maximum 3 years)

Closing Date: 27 November 2024 (at 11.59 pm) 

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