User talk:Nbarth/Projects

Questionable article
Hello, I saw the new article Categories of manifolds that you have created, but I have strong doubts that this topic is suitable for an article (as opposed to, say, a list, which already exists). It is overly generic, does not add much besides possible confusion, and will be difficult to maintain. I am not sure whether you watch it, but I've left some comments on the talk page, proposing that you delete the article, distributing any new content over specialized articles on complex manifolds, differentiable manifolds, etc. Best, Arcfrk (talk) 02:32, 25 November 2007 (UTC)


 * If you do delete it, it would be wise to move it to your user page space. Articles like it are very nice, but not always easy to find sources for.  Wikipedia frowns on (including in wikipedia) original research and personal synthesis, but such synthesis as in that article is incredibly useful.  If your mathematical journey is anything like mine, it seems likely that as you continue more and more of your private synthesis will have sources and it would be much easier to add it to articles if it were already nicely written, modulo a  or two. JackSchmidt 05:14, 2 December 2007 (UTC)

Cladistics
Hi Nbarth. Forgive for raising this directly on your User talk, but I find your recent change puzzling. I was hoping you might want to reply here since that page is occasionally contentious. Do you have any source for this comment? I believe that Wilhelm Johannsen's concept of the genotype only came in with the lab-based style of genetics of which Thomas Hunt Morgan was a major practitioner. Genotype versus phenotype is the type of situation where there are unexpressed qualities in the genome (such as recessive genes) which are only unveiled through breeding experiments and are otherwise not seen in the phenotype (the visible organism). From my reading, ideas about the genotype didn't influence evolutionary studies until the 1930s. Although Hennig was developing his system in the 1930s, there would have been little thought given to the genotype of organisms in the wild. I believe that Theodosius Dobzhansky was one of the first to study the distribution of lab-identified mutations in natural populations (and thus their genotype), and I'm not aware he influenced Hennig's thinking in any way. Thanks, EdJohnston (talk) 03:53, 7 March 2008 (UTC)


 * Discussion at User talk:EdJohnston.


 * Thanks for your last reply, which I had overlooked. That paper by Guenter Bechly is interesting, and it contains some references that would help bring to bring the Cladistics article up to date. This is a 150-year-old debate where everyone keeps using the same terms, but with different meaning, and they get indignant when others try to use different definitions of those terms. (Hennig appeared to think that Mayr was being perverse for not understanding obvious facts, and misconstruing terms that were well-understood, at least by him).  Good luck in your reform plans!  I had been thinking of writing about the Mayr-Hennig debate, since that would be a straightforward telling of history, and people would not expect to be able to changes to it without any sources. EdJohnston (talk) 17:16, 12 March 2008 (UTC)

Fitting series

 * In reply to your message

Howdy, your changes and new redirects look good. I think it is interesting that the lower Fitting series is refined by the lower central series(es), but the upper central series is harder to compare to the upper Fitting series. This is partially because a Fitting series has no relation to the ambient group, while a central series is only defined given some ambient group; while working downwards the ambient group coincidentally is visible to both, but while working upwards it is only visible to the central series.

People certainly break up finite solvable groups in very careful ways. For computational purposes the lower central series is not refined enough, and so instead one uses the lower exponent-p central series, where the ambient group centralizes each factor and each factor is elementary abelian. This stabilizes exactly where the lower central series stabilizes, and so refines the lower central series, and repeating in the same way, refines the lower Fitting series. However, doing this arbitrarily is not nice enough, as it only captures the horizontal layers of chief factors. To be more useful, it must also capture the vertical layers, the Sylow subgroups. A solvable group has a Sylow basis, and so the series is chosen very carefully to also exhibit the Sylow basis. The result is a composition series called a "special polycyclic generating system."

When dealing with infinite solvable groups, it is more common to just use abelian series. Instead of the Fitting subgroup, some people use the Hirsch-Plotkin radical—the unique largest locally nilpotent normal subgroup. The locally nilpotent normal (and subnormal) subgroups should form a complete sublattice. Schmidt's book on Subgroup lattices of groups is reasonably readable, and talks about a few pretty cool ideas; it goes slow with the lattices and fast with the group theory though, so some people find it frustratingly fast and slow at the same time. JackSchmidt (talk) 04:55, 7 March 2008 (UTC)