Dr David Craven MSci DPhil

Dr David Craven

School of Mathematics
Senior Birmingham Fellow

Contact details

Address
School of Mathematics
Watson Building
University of Birmingham
Edgbaston
Birmingham
B15 2TT
UK

Dr Craven is a Senior Birmingham Fellow in pure mathematics in the School of Mathematics, where he has been based since 2011.

His research interests are primarily in topics associated with the representation theory and subgroup structure of finite groups, and associated fields. In the past he has worked on representations of symmetric groups, infinite group theory, structure of group rings, and physical chemistry.

Personal website

Qualifications

  • DPhil in Mathematics, University of Oxford, 2008
  • MSci in Mathematical Sciences, University of Birmingham, 2004

Biography

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.

Teaching

Semester 1

Graduate course: Class field theory.

Semester 2

Graduate course: Representation theory of finite groups.

Postgraduate supervision

Dr Craven supervises PhD students in algebra, in fields such as group representation theory, finite and algebraic group theory, and fusion systems.

Research

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 four 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 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 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 myraid new exotic systems, together with examples of simple fusion systems exhibiting a variety of interesting behaviours.

Other activities

Editor, Beiträge zur Algebra und Geometrie, 2011--present.

Editor, London Mathematical Society journals, 2017-2021.

Royal Society grants panel member.

Member of the School of Mathematics communications team.

Publications

Recent publications

Book

Craven, D 2019, Representation Theory of Finite Groups: a Guidebook. Universitext, Springer Nature, Switzerland.

Article

Craven, D, Stewart, D & Thomas, A 2021, 'A new maximal subgroup of E8 in characteristic 3', Proceedings of the American Mathematical Society. https://doi.org/10.1090/proc/15759

Craven, D 2021, 'On medium-rank Lie primitive and maximal subgroups of exceptional groups of Lie type', Memoirs of the American Mathematical Society.

Rouquier, R, Craven, D & Dudas, O 2020, 'Brauer trees of unipotent blocks', Journal of the European Mathematical Society, vol. 22, no. 9, pp. 2821-2877. https://doi.org/10.4171/JEMS/978

Craven, D 2020, 'Trivial-source endotrivial modules for sporadic groups', Beitrage zur Algebra und Geometrie. https://doi.org/10.1007/s13366-020-00521-8

Craven, D 2019, 'Maximal PSL2 subgroups of exceptional groups of Lie type', Memoirs of the American Mathematical Society.

Craven, D 2018, 'Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic p', Journal of Group Theory, vol. 21, no. 5, pp. 719-787. https://doi.org/10.1515/jgth-2018-0014

Craven, D 2017, 'Alternating subgroups of exceptional groups of Lie type', London Mathematical Society. Proceedings , vol. 115, no. 3, pp. 449-501. https://doi.org/10.1112/plms.12043

Craven, D, Oliver, B & Semeraro, J 2017, 'Reduced fusion systems over p-groups with abelian subgroup of index p: II', Advances in Mathematics, vol. 322, pp. 201–268. https://doi.org/10.1016/j.aim.2017.10.001

Craven, D 2014, 'The Brauer trees of non-crystallographic groups of Lie type', Journal of Algebra, vol. 398, pp. 481-495. https://doi.org/10.1016/j.jalgebra.2013.06.002

Craven, DA & Pappas, P 2013, 'On the unit conjecture for supersoluble group algebras', Journal of Algebra, vol. 394, pp. 310-356. https://doi.org/10.1016/j.jalgebra.2013.07.014

Craven, DA & Rouquier, R 2013, 'Perverse equivalences and Brouéʼs conjecture', Advances in Mathematics, vol. 248, pp. 1-58. https://doi.org/10.1016/j.aim.2013.07.010

Craven, DA & Glesser, A 2012, 'Fusion systems on small p-groups', Transactions of the American Mathematical Society, vol. 364, no. 11, pp. 5945-5967. https://doi.org/10.1090/S0002-9947-2012-05580-2

Craven, D 2011, 'Algebraic modules and the Auslander–Reiten quiver', Journal of Pure and Applied Algebra, vol. 215, no. 3, pp. 221-231. https://doi.org/10.1016/j.jpaa.2010.04.013

Comment/debate

Craven, D 2015, 'The statistical sins of Jeremy Hunt', BMJ (Online), vol. 351, h6358. https://doi.org/10.1136/bmj.h6358

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