The main areas of Dr Craven's research activity are representation theory of finite groups and fusion systems.
In finite group representation theory, Professor Raphael Rouquier of Oxford and Dr Craven have embarked on an ambitious project to prove, or at least make substantial progress on, Broué’s abelian defect group conjecture, particular for principal blocks of finite groups of Lie type. In recent work they have laid the foundations of a systematic attack on the geometric form of Broué’s conjecture, using the new concept of perverse equivalences.
In fusion systems, Dr Craven focuses on the algebraic side, attempting to construct an internal theory of fusion systems, that neither translates bolidy results from local finite group theory nor relies heavily on topological intuition, the two currently most successful methods of approaching the subject. This approach manifests itself in his recent theorem proving the equivalence of the two definitions of a simple fusion system.