This is sort of a word of explanation as to why I haven’t been as active in the past week or two as I had previously been, and a bit of a preview of my plans for the blog in the near future.

First, my research project on counting acyclic orientations has been moving along, but I don’t quite feel comfortable (yet) posting my progress on this blog. The good news is that there seems to be a promising path to an asymptotic formula for at least a toy case; unfortunately, there have been other demands on my time lately and I haven’t been able to set aside a few hours to pursue it. Once I have that chance, I’ll probably say a little more about what’s happening on that front.

In other news: my post on category theory and combinatorics has gotten a good deal of (what I consider to be) positive feedback, which of course I’m very happy about. Unfortunately, I’m still pretty much a total novice to the categorical point of view, so I haven’t yet processed everything that’s been said in the comments thread. At some point, I’m hoping to write a follow-up post in which I try to expand upon a couple of the points that have come from that, but it’s not going to happen anytime particularly soon.

I also have a few draft articles for the blog, in various stages of completion. One grew out of my comment about exceptional structures and has to do with mathematical beauty. It’s pretty philosophical and I’m likely to just end up embarrassing myself, but I nevertheless plan to post it once I’m inspired to finish it.

Another post is nearly done, but I’m consulting an expert first to make sure that I don’t *totally* end up with egg on my face. (Not so “uninformed” anymore! Has PotM jumped the shark? Discuss.) Once I hear back from him, or I decide it’s a lost cause, we’ll probably be good to go.

Apart from that, I have two non-mathematical (well, semi-) posts, one of which I’ll probably finish and one of which I probably won’t, and a couple of crazy crackpot ideas that it turns out probably won’t work, and which I’ll either delete or save for a “Harrison’s Craziest Crackpot Ideas” special edition.

Last but certainly not least: I’m thinking of starting an actual series of posts aimed at a more general audience, and in which I basically go through a standard first-semester undergraduate algebra course from a more intuitive and discovery-oriented perspective than normal. There are a few pitfalls here, of course: first of all, I’m not an algebraist, I’ve never actually even taken a real algebra course for credit. (Honestly, this is something I’m doing as much for my own benefit as for anyone else’s.) Second, and probably more importantly, I don’t have much of a curriculum mapped out right now. I’d like to (try to) motivate the definition of a group, and probably prove some basics like Cayley’s and Lagrange’s theorems, but that’s one, maybe two posts, and beyond that I’m more-or-less in the dark pedagogically. (This probably has something to do with the fact that I’ve never taken undergraduate algebra.) Does anyone have any suggestions as to where I could go from there? I’d like to get into ring theory, of course, and maybe even a little bit of field theory and Galois stuff, but I don’t know what’s usually taught (or, sometimes, how best to motivate it.)

Anyway, with any luck there’ll be a new post sometime in the next couple of days. My summer classes end in two weeks, and I’m going on vacation after that, but at the very latest I should be back on schedule by early August.

Tags: non-math

July 16, 2009 at 04:31 |

Have you read through the Unapologetic Mathematician’s category theory-oriented approach to group theory? There are interesting ideas there. It might also be worth, for example, reading through Ash’s Abstract Algebra, which is freely available online. It’s reasonably compact.

July 16, 2009 at 05:11 |

Heh, Ash is where I actually learned most of what algebra I know. Part of the problem, though, is that it’s at least nominally a “graduate” course, and it’s not really in the sort of intuitive style that I’m aiming for. (I don’t want to use any word ending in “orphism” in any of the lectures.) Still, yeah, it’ll probably be a major reference for when I’m writing the posts.

I haven’t seen John Armstrong’s group theory posts — I’ll definitely take a look.

July 17, 2009 at 17:28 |

“I’m thinking of starting an actual series of posts aimed at a more general audience, and in which I basically go through a standard first-semester undergraduate algebra course from a more intuitive and discovery-oriented perspective than normal. There are a few pitfalls here, of course: first of all, I’m not an algebraist, I’ve never actually even taken a real algebra course for credit. (Honestly, this is something I’m doing as much for my own benefit as for anyone else’s.)”

I’ve been doing a similar thing on Lie algebras (starting here http://deltaepsilons.wordpress.com/2009/07/16/lie-algebras-fundamentals/#more-54, but to be continued definitely) at the Delta Epsilons group blog to cover, I guess, the basics of representation theory. I really need to know this material as well as possible, which is a large part of my motivation in writing these posts. I’ve never taken a general algebra course either (or one on representation theory) so I’m learning quite a bit from the comments. My own exposition will largely be derived from books that I am reading on this subject and conversations with various people.

Incidentally, you didn’t mention Sylow’s theorem as a possible topic, which could take a few posts to cover the many different proofs (though perhaps you already intended it). Also, principal ideal domains and unique factorization, localization of rings, and Hilbert’s basis theorem could furnish a few more (perhaps non-standard) posts.