tag:blogger.com,1999:blog-8666091.post116509455848355788..comments2021-05-03T21:54:48.969+02:00Comments on The Reference Frame: Monstrous moonshine, finite groups, and string theoryLuboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-8666091.post-1165178121007894992006-12-03T21:35:00.000+01:002006-12-03T21:35:00.000+01:00it also defines the densest packing in 24 dimensio...<I>it also defines the densest packing in 24 dimensions.</I><BR/><BR/>And it is also one if the highest known regular densities independently of dimension. See the table plot of page 6 of <A HREF="http://arxiv.org/pdf/math.CO/0207256" REL="nofollow">math.CO/0207256</A>. Only the lattice P48q, in 48 dimensions, gets a better density.<BR/><BR/>btw, as some of the readership of your blog does not intersect with s.p.s, let me repeat here the question I did in October: is there any use of this fact to optimise pattern recognision in neural networks? It could be a very interesting biomathematical result if it were the case that information (speech) can be better transmited and recognised by using a set of about 24 patterns.Leucipohttps://www.blogger.com/profile/14505549871207858030noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1165176830958758402006-12-03T21:13:00.000+01:002006-12-03T21:13:00.000+01:00Lately, "essentially the monster group" has been t...Lately, "essentially the monster group" has been translated to the name "Fake Monster Lie Algebra".<BR/><BR/>It should be mentioned that a lot of the work of Borcherds is also uploaded to the ArXiV. Besides the Leech lattice, a lot of work in Borcherds' thesis was done in its cousin on 25+1 dimensions, as a lot of the properties of lattices depend only on signature.<BR/><BR/>The 8-periodicity of the selfdual unimodular lattices seems very much as the one of the spinorial clock, but I have not seen this Clifford-algebraic setting in a explicit way. Perhaps because Conway conjectured that it was more important to look for arguments to finish the series in signature 24, so the important objects seem to be your aformentioned \Gamma 8 and \Gamma 16, and then the famili of lattice in euclidean dimension 24 (or minkowski 26). But it could be interesting to work out explicitly the relationship with Clifford algebras and then with Bott periodicity. I could even volunteer to collaborate on it: my motivation is that the proof of existence is done via a theorem of Milgram, quotienting the dual lattice by the original one, which in the self-dual case produces a trivial object, and I am intrigued about what happens if instead of the naive quotient we use the way of crossed products (Connes + Morita equivalence methods).Leucipohttps://www.blogger.com/profile/14505549871207858030noreply@blogger.com