2-representation theory and applications to representation theory and number theory (MIEMIETZ_U27SCI)
Key details
- Application Deadline
- 24 February 2027 (midnight UK time)
- Location
- UEA
- Funding Type
- Competition Funded Project (Home students only)
- Start Date
- 1 October 2027
- Mode of Study
- Full-time
- Programme Type
- PhD
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Project Description
Primary supervisor - Prof Vanessa Miemietz (opens in a new window)
In the last 20 years, categorification has started to play an important role in representation theory. 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, as well as counterexamples to James’ conjecture and Lusztig’s conjecture.
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. On the other hand, the natural transformations often form algebras that we are already interested in and we obtain categorical actions on their representation categories.
The most important examples include:
1) categorifications of quantum groups, where applications are obtained by analysing the action of these categorified quantum groups on suitable (derived) categories of representation of p-adic general linear groups with the aim to translate this information through the local Langlands correspondence to obtain new information on questions in number theory.
2) categorifications of Hecke algebras via Soergel bimodules, which have very interesting and only partially understood 2-representation theory in their own right, as well as applications to the representation theory of algebraic groups through studying the categorification of the anti-spherical module.
This PhD project will study some of these examples, for which it will also be necessary to develop the abstract theory further.
Entry Requirements
The standard minimum entry requirement is 2:1 in Mathematics.
Funding
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 for 3 years. The current UEA / institutional PhD stipend rate is £21,805 and will be subject to any agreed inflationary uplift for 2027/28.