The product rule of the all familiar operation of taking derivatives of real valued functions has a plethora of generalisations and applications in algebra. It leads to the notion of derivations of algebras - these are linear endomorphisms of an algebra satifying the product rule. They represent the classes of the first Hochschild cohomology of an algebra. The first Hochschild cohomology of an algebra turns out to be a Lie algebra, and more precisely, a restricted Lie algebra if the underlying base ring is a field of positive characteristic. The (restricted) Lie algebra structure extends to all positive degrees in Hochschild cohomology - this goes back to pioneering work of Gerstenhaber on defornations of algebras. Modular representation theory of finite groups seeks to understand the connections between the structure of finite groups and the associated group algebras. Many of the conjectures that drive this area are - to date mysterious - numerical coincidences relating invariants of finite group algebras to invariants of the underlying groups. The sophisticated cohomological technology hinted at in the previous paragraph is expected to yield some insight regarding these coincidences, and the present proposal puts the focus on some precise and unexplored invariance properties of certain groups of integrable derivations under Morita, derived, or stable equivalences between indecomposable algebra factors of finite group algebras, their character theory, their automorphism groups, and the local structure of finite groups.

Doctoral Training Partnerships: a range of postgraduate training is funded by the Research Councils. For information on current funding routes, see the common terminology at https://www.ukri.org/apply-for-funding/how-we-fund-studentships/. Training grants may be to one organisation or to a consortia of research organisations. This portal will show the lead organisation only.

further details to be added at supervisor

Doctoral Training Partnerships: a range of postgraduate training is funded by the Research Councils. For information on current funding routes, see the common terminology at https://www.ukri.org/apply-for-funding/how-we-fund-studentships/. Training grants may be to one organisation or to a consortia of research organisations. This portal will show the lead organisation only.