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Categorified representation theory


   School of Mathematics

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  Dr V Miemietz  No more applications being accepted  Self-Funded PhD Students Only

About the Project

This PhD project will investigate questions surrounding categorified representation theory.  

Categorification has led to many breakthroughs in representation theory in the last 15 years. On the most basic level (going in some sense from categorical level zero to level 1), vector spaces over a fixed field categorify natural numbers. By taking the dimension of a vector space, we can reproduce a natural number, but we have additional information for vector spaces when looking at linear maps from one to the other that are not visible purely on the level of dimensions. In representation theory, one typically looks at passing from categorical level one to categorical level 2, and roughly speaking, replaces algebras acting on vector spaces (in classical representation theory) by higher categorical objects, more specifically 2-categories, acting on categories. This gives rise to new methods to tackle long-standing problems and has led for example led to a proof of Broué's abelian defect group conjecture for symmetric groups, an algebraic proof of the Kazhdan--Lusztig conjectures for all Coxeter types, and counterexamples to James’ conjecture.

The PhD project will focus on both advancing the general theory and studying special classes of examples of categorification, e.g. coming from Soergel bimodules, which categorify Hecke algebras. The technical ingredients for the project will come from classical representation theory of finite-dimensional algebras, category theory and homological algebra. 


Funding Notes

This PhD project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found at https://www.uea.ac.uk/about/university-information/finance-and-procurement/finance-information-for-students/tuition-fees
A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. Applicants should contact the primary supervisor for further information about the fee associated with the project.

References

i) V. Mazorchuk, Lectures on algebraic categorification. QGM Master Class Series. European Mathematical Society (EMS), Zürich, 2012.
ii) V.Mazorchuk, Classification problems in 2-representation theory. São Paulo J. Math. Sci. 11 (2017), no. 1, 1–22.
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