Solution of Broue's conjecture in representation theory of finite groups

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The head investigator Shigeo Koshitani has gotten the following results in the last three yeas.Firstly, Koshitani worked with Morton E.Harris (University of Illinois at Chicago, and we proved that there is a Morita equivalence between two block algebras, say A and B such that A and B are corresponding via the Glauberman corresondence and moreover, the Morita equivalence is described explicitly by an (A,B)-bimodule M which induces the original Glauberman correspondence at the level of ordinary characters. This result has been published in the Journal of Algebra (Elsevier).Secondly, Koshitani has been working with Naoko Kunugi and Katsushi Waki together, in order to solved Broue's abelian defect group conjecture for the fourth Janko simple group J4. This group J4 is relatively quite large in the twenty-six sporadic simple groups. The three people above proved that Broue's abelian defect group conjecture is true for the group J4 for all prime numbers. This result was published in the Jour … More nal of Pure and Applied Algebra (Elsevier). Koshitani has been working on also another kind of Broue's conjecture. Namely, he has been looking at blocks of finite groups whose defect groups are not abelian but metacyclic groups. He with other two co-researchers has gotten some results on the subject as well.The head investigator Shigeo Koshiani announced and gave talks on the above results in many places. Such as at the Mathematical Institute of Oberwolfach, Germany (2006), the Mathematics Luminy Institute, France (2007),University of Chicago, Mathematical Institute of the University of Oxford, University of Aberdeen in the UK, University of Leeds in the UK, University of Jena, Germany, University of Braunsweich Technology, Germany, National University of Ireland in Maynooth, Kyoto University, and so on.On the other hand, Naoko Kunugi, who has been one of the investigators of this project also gave several invited talks on Broue's abelian defect group conjecture. For instance, she gave two talks in the Symposium on algebraic groups and quantum groups held in May 2006 near Nagoya, Japan, and also she gave an invited special talk at the annual meeting orthorized by the Mathematical Society of Japan in September 2006 in Osaka City University. Less