Today was the last day of a five-day long conference on group theory. This was the Glauberman conference, held right here at the Mathematics Department of the University of Chicago. The conference was in honor of Professor George Glauberman, a leading group theorist who’s proved results like the ZJ-theorem and the Z*-theorem. Prior to this conference, I’d heard of mathematical conferences and read books with proceedings of these conferences, but I didn’t have any experience of attending a conference. So I was very eager to attend this one. “Conference” can have many meanings. The Glauberman conference was primarily a series of lectures by different mathematicians on different topics. In fact, most of the lectures were on very specific topics, they were short (about 30-35 minutes) and there wasn’t a unifying theme to the talks. I didn’t follow too much of the content of the talks, primarily because of the fast pace and the large number of talks. But it was a great experience to meet people from across the world (group theorists from Europe and Japan had also come). I’d read books writen by some of these people, and had also corresponded with some of them, so it was nice to see them in person (though the schedule was too hectic to interact more with them). I got an idea of the notational conventions that were followed in group theory. I learned that the default convention in group theory is to make elements act on the right (especially when writing down cumbersome commutators and expressions to simplify) rather than on the left. More importantly, I got to understand some of the important research themes in the subject. One research theme is around a collection of conjectures intended to understand better the relation between the representation theory of a huge group, and the representation theory of “local” subgroups (small subgroups that occur as normalizers of subgroups of prime power order). The first conjecture in this regard was by McKay. McKay conjectured that the number of irreducible representations of order of a group equals the number of irreducible representations of order of the normalizer of any -Sylow subgroup. Many modifications of this conjecture have been proposed by Alperin, Isaacs and others. In a similar but somewhat different vein, there’s the Glauberman correspondence, that gives an explicit bijection between the representations of a huge group and a smaller subgroup. This, too, has spawned a number of related thoughts. There were some talks in the Glauberman conference that focused on some of the applications and results inspired. Professor Bhama Srinivasan, who gave a lecture about some correspondences involving linear groups, told us that the whole spectrum of conjectures had been summarized as “I AM DRUNK” where the letters stood for the initials of the people who had come up with variations of the McKay conjecture. (I, A, M stand for Isaacs, Alperin and McKay; I forget all the other letters right now). Another important theme was the theory of “replacement”: replacing a subgroup satisfying certain, weaker properties, with a subgroup satisfying certain, stronger properties. Thompson was the first person to come up with replacement theorems, and Professor Glauberman has published a number of recent results in that regard, making good use of the ideas behind the Lazard correspondence. One of the interesting results was mentioned by Professor Khukhro, who was inspired by Professor Glauberman’s replacement theorem to prove a very general result that works for all groups: a normal subgroup of finite index can be replaced by a characteristic subgroup of finite index, and satisfying the same multilinear commutator identity (so for instance a normal nilpotent subgroup of finite index can be replaced by a characteristic nilpotent subgroup of finite index). Group theory’s recently winning the attention of people in topology and category theory. During the classification of finite simple groups, there were some “candidates” for finite simple groups that never materialized into actual groups. however, there was a lot of data in these cases to suggest that a group exists. later, it was discovered that one could define an abstract notion, called a fusion system, and that every group gives rise to a fusion system, but there are fusion systems that don’t come from groups. Fusion systems are something like a piece of consistent data that could have come from a group, but on the other hand, may not. Some recent work has gone on into find out what are the fusion systems that do not come from groups, and how one can judge whether a given fusion system arises from a group. The talks at the Glauberman conference weren’t directly on these basic concerns, but on some related research. This included talks by Cermak, Bob Oliver, and Radha Kessar. There were also talks related to classifying and making sense of the -groups (groups whose order is a power of a prime). Classifying groups is a tricky proposition: it makes sense only if we decide what it means to “classify. Professor Leedham-Green gave a talk on classifying -groups by coclass. All the talks were 30-35 minutes long. Some of them used the chalkboard, others used laptop-based presentations, and yet others used transparencies. In fact, the ones using transparencies used two projectors, with one projector used to show the “previous” transparency for reference. It was good fun.
March 28, 2008