PhD Studentship: Representations of P-Local Finite Groups

The theory of p-local Finite Groups is analogous to that of Finite Groups, and provides a unifying approach to classification problems in algebra and topology. One family of “atoms” of p-local symmetry – the p-local Finite Groups of Lie type – are a natural family of objects for which one can formulate and prove local-global conjectures generalising analogous statements in the modular representation theory of Finite Groups. This suggests a structured approach to these conjectures exists, but an incomplete understanding of the nature of a “representation” of a p-local finite group of Lie-type is currently holding back progress in the area. Our project proposes vector bundles over these spaces as the correct alternatives to representations. As a useful test case, we will construct irreducible vector bundles for the smallest class of objects in this family - the Sullivan spheres – and show that their degrees match those predicted by the associated cyclotomic Hecke algebras, thus unifying apparently disparate objects in algebra and topology. Representation theory is a key interest of the Mathematics Department’s Geometry and Mathematical Physics research group, and you will be part of a team of notable pure mathematicians working in related areas including Sasha Vesolov and Jenya Ferapontov who have been professors at Loughborough since 1995 and 2008 respectively. The PhD student will be supported through joint weekly research meetings and trained on technical skills such as drafting publications, writing code in various mathematical packages including (but not limited to) MAGMA/GAP/Sage/Python and presenting at international conferences. There will be opportunities to travel nationally and internationally. We actively encourage women, disabled, Black, Asian and Minority Ethnic candidates and LGBTQ+ candidates, who can bring their experiences and voices to the team/studentship.