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My main focus at the moment is on Donovan's conjecture and the classification of Morita equivalence classes of blocks of finite groups. Because of work with Kessar, Külshammer and Sambale, Donovan's conjecture is known for 2-blocks with elementary abelian defect groups, that is, there are only finitely many Morita equivalence classes of such blocks. This opens the question of classifying the Morita equivalence classes. This research was funded by EPSRC grant Morita equivalence classes of blocks.
An important tool in classifying Morita equivalence classes, and an interesting invariant in its own right, is the Picard group. Research into Picard groups, amongst other things, continued in the EPSRC funded grant Representation theory over local rings, which was a joint project with Radha Kessar at City, University of London.
I have set up this wiki site for cataloguing blocks of finite groups up to Morita equivalence and recording progress on Donovan's conjecture. If you would like to contribute, then please let me know.
Claudio Marchi (completed 2022)
Cesare Ardito (completed 2020) Blocks with an elementary abelian defect group in characteristic two
Elliot Mckernon (completed 2020) 2-Blocks with homocyclic defect group whose inertial quotient contains a Singer cycle
Inga Schwabrow (completed 2016) The centre of a block
Pornrat Ruengrot (completed 2011) Perfect isometry groups for blocks of finite groups
Stavros Apostolou (completed 2009) Generalisations of the representation theory of p-solvable groups
Blocks whose defect groups are Suzuki 2-groups, arXiv 2401.04028
(with M. Livesey) Morita equivalence classes of 2-blocks with abelian defect groups of rank 4, to appear, J. LMS
(with J. An) Morita equivalence classes of blocks with extraspecial defect groups p_+^{1+2}, arXiv 2310.02150
(with M. Livesey) Donovan's conjecture and extensions by the centralizer of a defect group, J. Algebra 582 (2021), 157-176.
(with M. Livesey) Some examples of Picard groups of blocks, J. Algebra 582 (2021), 157-176.
(with F. Eisele and M. Livesey) Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings, Math. Z. 295 (2020), 249-264.
Morita equivalence classes of blocks with elementary abelian defect groups of order 16, arXiv 1612.03485
(with M. Livesey) Donovan's conjecture and blocks with abelian defect groups, Proc. AMS. 147 (2019), 963-970.
(with M. Livesey) Towards Donovan's conjecture for abelian defect groups, arXiv 1711.05357, J. Algebra 519 (2019), 39-61.
(with M. Livesey) Classifying blocks with abelian defect groups of rank 3 for the prime 2, J. Algebra 515 (2018), 1-18.
(with M. Livesey) Loewy lengths of blocks with abelian defect groups, Proc. AMS Ser. B 4 (2017), 21-30
Morita equivalence classes of $2$-blocks of defect three, Proc. AMS 144 (2016), 1961-1970
(with R. Kessar, B. Külshammer and B. Sambale) $2$-blocks with abelian defect groups, Adv. Math. 254 (2014), 706-735
(with A. Moreto) Extending Brauer's height zero conjecture to blocks with nonabelian defect groups, Int. Math. Res. Not. 2014 (2014), 5581-5601.(available electronically)