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• DPhil in Mathematics, University of Oxford, 2008
• MSci in Mathematical Sciences, University Â鶹¾«Ñ¡, 2004

Dr Craven received an MSci from Birmingham in 2004, then moved to Oxford, where he was first awarded a DPhil from St John’s College in 2008, and then moved to Christ Church as a Junior Research Fellow for the next three years.

He was appointed as a Birmingham Fellow in November of 2011, and is now a Senior Birmingham Fellow. Between 2012 and 2021 he was a Royal Society University Research Fellow.

Semester 1

LH/LM Group theory

Semester 2

LH/LM Group theory

Research Themes

• Modular representation theory: particularly Broué’s conjecture and representations of finite simple groups
• Fusion systems
• Group rings of torsion-free groups
• Subgroup structure of exceptional groups
• Representation theory of symmetric groups and associated combinatorics 

Research Activity

The main areas of Dr Craven's current research activity are subgroups of simple groups, and representation theory of finite groups. In the recent past he has worked on fusion systems, and before that on a number of different areas.

Dr Craven's research in subgroup structure centres around the problem of classifying the maximal subgroups of the finite exceptional groups of Lie type. In a series of long papers, he has completed the classification for the series F₄, E₆ and ²E₆ in 2021. For E₇, in more recent work he has almost completed the classification, with just three outstanding candidate subgroups. Work on E₈ is not as far advanced, and there is still much work to be done in this area.

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 bodily 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 one theorem proving the equivalence of the two definitions of a simple fusion system. More recently, with Bob Oliver and Jason Semeraro, he completed a classification of a particular class of fusion system, which throws up myriad new exotic systems, together with examples of simple fusion systems exhibiting a variety of interesting behaviours.

• Editor, Beiträge zur Algebra und Geometrie, 2011-present.
• Editor, London Mathematical Society journals, 2017-2021.