## Participant Testimonials

Mar 16 - Mar 21, 2014

It was a wonderful experience to be at BIRS. The following developments would not have happened (at all or as quickly) without the visit to Banff: 1) I gave a talk on "p-permutation equivalences", a relatively new concept in the theory of blocks of finite group algebras. After the talk, Jürgen Müller and Susanne Danz approached me with the suggestion to write a computer program that would allow to compute and analyze p-permutation equivalences and more generally p-permutation modules. We met for about 3 hours, discussed possible algorithms, and have now an initial concept. Motivated by this quick progress we agreed to meet for a week in July to continue with the project. Computations of specific examples would be very enlightening, especially in order to settle or provide more evidence on uniqueness questions. 2) There is a logical hierarchy between different types of equivalences of blocks of finite groups. Discussions with Jeremy Rickard and Markus Linckelmann gave me new ideas how to construct p-permutation equivalences that do not come from the stronger notion of a splendid Rickard equivalence. Examples of this type are currently unknown. 3) In collaboration with Susanne Danz we continued to complete a paper that we started more than a year ago on the quasi-heredity of certain category algebras. "Skype" or similar media would have been far less efficient for this purpose.

Mathematics, University of California Santa Cruz

This meeting was a marvelous success and I enjoyed every minute of it. I guess much has been said about the comfort of the venue and the beauty of the surroundings. It might be added that the new auditorium is very intelligently conceived and that the recorded talks were very appreciated. For me the event was a unique opportunity to discuss the maths I am presently working on, having focused on this theme in the last three years. I particularly enjoyed the very interesting and carefully prepared session about the remaining problems in checking for simple groups of Lie type the inductive conditions available for certain global/local conjectures. In all it was a very fruitful meeting due to the presence of almost all experts of the field, plus a couple of other experts in neighboring subjects, like Ben Elias whose talk and enthusiasm I appreciated very much. It is also very positive that this meeting could invite many junior researchers (post doc and PhD students). I had a series of informal discussions with many participants. With Tiep I discussed the possibility of proving certain stabilizer statements in the case of finite symplectic groups from the existing literature he knows well. With Brunat, following Gramain's talk on their joint work, I discussed the possible applications of the representation theory of wreath products and the related Morita equivalences to the proof of more equivalences between blocks of Weyl groups or Hecke algebras of classical types. I had many discussions with Paul Fong about the relevance of the joint work with Broue and Srinivasan on Dade's conjecture in the case of unipotent blocks of finite reductive groups, especially in view of recent results by Spaeth producing a reduction theorem for that conjecture.

Marc Cabanes Universite Paris Diderot

I found the workshop at BIRS very inspiring. Besides many interesting talks on recent results and trends in global-local representation theory of finite groups, the meeting also provided ample opportunities for personal discussions with colleagues from all over the world. Here are some concrete research activities resulting from the workshop at BIRS: Inspired by Robert Boltje's (UC Santa Cruz, USA) talk on Tuesday, he, Jürgen Müller (University of Jena, Germany) and I started to discuss possible strategies for investigating $p$-permutation equivalences between blocks of finite groups computationally. During the problem session on Thursday afternoon we developed some concrete ideas, and we now plan to meet again this summer to pursue our joint project. Moreover, during the free afternoon on Wednesday, Robert Boltje and I had the chance to make significant progress on another joint project (on quasi-hereditary twisted category algebras), which we had started several months ago, but were not able to complete just via email or Skype correspondence.

Susanne Danz Department of Mathematics, University of Kaiserslautern

I found the workshop at BIRS very stimulating. Meeting all the renowned experts in the field was a very interesting and motivating experience. My research background is only tangential to the topic of the workshop, but I am eager to learn more about this topic and I will continue my research in this direction. (More precisely, I am working on classifications of objects which are vital for the local/global conjectures discussed in the workshop.) I also bought two books, written by participants of the workshop, and I will do more research in this field. The stay was wonderful! Thank you very much! I am looking forward to coming back in November!

Dr
School of Mathematical Sciences, Monash University

I enormously enjoyed my stay at BIRS; the workshop was very inspiring for me. Besides getting a great overview on current developments and meeting and discussing with people whom I haven't seen for a while, I found particularly useful the problem session/discussion, in which a variety of problems (from major directions to important obstacles of a mere technical nature) were explained and possible ways discussed on how to overcome them. This certainly provided some fresh insights for me which -- I hope -- may even lead to some new research projects and collaborations.

Lehrstuhl für Algebra, Universität Stuttgart

The conference was a great success; it was excellently prepared and managed by the organizers and the people at BIRS. Most of the talks reported on recent progress on conjectures by Alperin, Brauer, Broué and McKay. It was exciting to see that we are really coming closer to the solution of some of these longstanding open problems. There was also sufficient time for exchange and cooperation with the other participants. I personally profited, in particular, from discussions with M. Geline, M. Linckelmann and A. Turull on vertices of simple modules and characters. R. Kessar, M. Linckelmann and I are in the course of preparation of a manuscript on this topic which will hopefully be available in a few weeks or perhaps months. We are all very grateful to the BIRS for providing such excellent facilities for mathematics and hope to return some time in the future.

Mathematical Institute, University of Jena

[As a result of the workshop], I have started a new line of research with M. Linckelmann about characterizing nilpotent blocks, which is a possible continuation of one of my recent papers. Also, with my coorganizer P. H. Tiep, we have started a possible refinement of McKay in groups with nilpotent Hall subgroups. About my student C. Vallejo, she gave her first important international talk, and she already has an invitation to visit Kaiserlautern with Gunter Malle for some months to join his team working on the McKay conjecture. The talks were, in general, excellent. The discussion session on the route plan to prove the global/local conjectures was very useful for researchers, to know where we are exactly, and for younger researchers (and even senior ones) to join us in the plan of proving them.

Gabriel Navarro Professor
Algebra, University of Valencia

I want to thank the organizers once again for a very nice workshop and the opportunity to speak. It was very beneficial to be able to meet (or see again) many people in the field, both those who are well-known in the area and also fellow early-career mathematicians. Following my talk, there were some very good recommendations and discussions, for example with G. Malle, G. Navarro, and J. Taylor, which may help my progress on one of my current problems. I also had the opportunity to speak with G. Malle and P.H. Tiep about future directions for my research involving the inductive Alperin-McKay and BAWC conditions, as well as S. Koshitani about new ways to simplify some arguments in my earlier work on the inductive conditions, which will be quite helpful in my future research.

Mathematics, Michigan State University

The workshop on Global-Local Conjectures in Representation Theory of Finite Groups was very successful. The participants were a good mix of senior and junior mathematicians, and there were lectures by both. I had discussions with several participants. One lecture in particular by Olivier Dudas on work he was doing with Gunter Malle (an organizer) gave me insight into my research and leads I can pursue. Among new contacts I would specially mention Ben Elias, a young mathematician who is doing cutting edge research. On the final day Gunter Malle gave an overview of where the program on Global-Local Conjectures stands, which would be very useful for both people working in that area now and for those interested in starting to work in it.

Mathematics, University of Illinois at Chicago

I would like to thank BIRS for giving us this opportunity to hold the workshop at the excellent venue and in the highly stimulating environment of the Banff International Research Station! During my stay at BIRS, aside from organizational tasks and attending interesting lectures and brainstorming discussion session, I was able to start a new research project with my collaborator Gabriel Navarro, as well as continue our collaboration on ongoing projects that aim to resolve some basic questions on representations of finite groups. Together with Gunter Malle and Frank Luebeck (and in discussion with Jean-Baptiste Gramain and Olivier Brunat), I had started another new research project with the aim to answer a question of R. M. Guralnick concerning representations of finite quasisimple groups. I also had discussions with other participants, including Marc Cabanes, Gerhard Hiss, Klaus Lux, Bhama Srivasan, and Amanda Schaeffer Fry.

Department of Mathematics, University of Arizona

Although this was not the first time I was invited to BIRS, it was the first time I was able to come. I was very impressed by the organization, and how well the facilities were administrated. The scientific aspects of the conferences were likewise excellent. The topics of the conference had a clear focus on the proof of a number of global/local conjectures in the representation theory of finite groups. The participants are making important and related progress on these, and exchanging ideas and divvying the work to do next is essential. I was particularly interested in learning the current status of these conjectures, including some new ones. I am pleased that there is substantial progress on many of the simple versions of the conjectures. At the same time, there is still a lot of work to do on the more refined versions of them. While there, I enjoyed productive mathematical discussion on these and related topics with a number of researchers, including Michael Geline, David Gluck, and Burkhard Kuelshammer. These and other informal exchanges affected a number of papers on these conjectures that I am in the process of writing, and suggested new promising avenues of research. It was an excellent and productive experience all around.

Mathematics, University of Florida