Modular representation theory: particularly Broué’s conjecture and representations of finite simple groups
The Unit Conjecture for group rings
Profinite and polycyclic groups
Representation theory of symmetric groups and associated combinatorics
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.
An up-to-date list may be found at web.mat.bham.ac.uk/D.A.Craven/papers.html
Craven, D.A. (2011), Normal Subsystems of Fusion Systems, J. Lond. Math. Soc., 84:137-158.
Craven, D.A., Eaton, C., Kessar, R., and Linckelmann, M. (2011), Blocks with a Klein Four Defect Group, Math. Z., 268:441--476.
Craven, D.A. (2011), Algebraic Modules and the Auslander–Reiten Quiver, J. Pure Appl. Algebra, 215:221–231.
Craven, D.A. (2010), Lower Bounds for Representation Growth, J. Group Theory, 13:873–890.
Craven, D.A. (2010), Control of Fusion and Solubility in Fusion Systems, J. Algebra, 323:2429--2448.
Craven, D.A. (2009), Simple Modules for Groups with Abelian Sylow 2-subgroup Are Algebraic, J. Algebra, 321:1473–1479.
Holloway, A.F., Craven, D.A., Xiao, L., del Campo, J. And Wildgoose, G. (2008), Developing Random Network Theory for Carbon Nanotube Modified Electrode Voltammetry: Introduction and Application to Estimating the Potential Drop between MWCNT-MWCNT Contacts, J. Phys. Chem. C, 112:13729–13738.
Craven, D.A. (2008), Symmetric Group Character Degrees and Hook Numbers, Proc. Lond. Math. Soc., 96:26–50.