What Is Research?

February 4, 2008

Quarterly progress

When I started life this quarter, I had determined that it would be more enjoyable than last quarter, with less paranoia about assignments, more fun in the learning process and a cooler and calmer perspective to life. Things have been going fairly well in all respects.

Probably the first difference is that I’m much calmer about assignments, even when they don’t get done or are left right for the last minute. Providence also seems to have helped me; the assignments are (by and large) shorter, though there are some exceptions and I have to sometimes do a hasty last-minute job. But then, I had to do hasty jobs last quarter too; the difference was that assignments occupied much more mental space so that I couldn’t concentrate on doing the things that I liked.

One thing I’ve been experimenting with is wikiing while I work, and that means that as I’m learning stuff, I’m constantly thinking of how it can be organized on and integrated with the wikis that I’m working on. I’ve been augmenting the Commutative algebra wiki as I go along. This hasn’t been instant magic, because I don’t have the kind of feel for commutative algebra to immediately see how certain facts can be organized, but it means that I’m thinking of the subject in a way that’s not just limited to assignments. The wiki’s also becoming a useful, no-nonsense, reference point for me, and a convenient way to augment my memory and intelligence.

Differential topology is very interesting, and while studying it, I have to keep updating two wikis, the Topology Wiki, and the Differential geometry wiki, which often cover similar stuff from slightly different perspectives. I had worked quite a bit on organizing the topology wiki over the winter so every new thing I want to say seems to have a nice place to put it, and it seems to be not too far when the wiki will start exhibiting the kind of beautiful self-organization that I’m seeing in the group theory wiki.

The fact that there’s a course on local analysis in finite groups keeps me very happy. Although I can’t devote too much time to group theory while in the midst of all my compulsory courses, this course at least keeps me on track in the subject. It’s fascinating to see in formal proof all the things that I have picked up from textbooks and miscellaneous papers. I hope that I can really work out on the Group Properties Wiki soon, though I keep augmenting it from time to time. It’s looking more and more beautiful.

I’ve also been discussing some ideas in group theory with Professor George Glauberman, the instructor for the course on finite groups. Again, I plan to pursue them more later on.

The functional analysis course isn’t going as well as I’d hope, but I’m still having fun trying to follow the ideas. The problem for me is that the topics and direction are changing rapidly. Assignment-solving had a huge collective component last week (in other words, for many problems, I couldn’t figure out the solutions even after I wrote them). But at least there are some things in the subject that I’m learning. I am trying to wiki things out there on the measure theory wiki, but since I hadn’t set it up and structured it, what I add ar ejust isolated articles, and there’s no bigger picture emerging.

There’s also a course by Professor Victor Ginzburg on semisimple groups and geometry that I’m attending. For the first time, I’m seeing proofs (although more on the line of outlines of proofs) for statements on algebraic groups. This isn’t my primary focus area but it is something I’d like to understand well and Professor Ginzburg’s approach is interesting and his excitement is infectious. Unfortunately, I’m not getting to spend time on this outside class. It does remind me of some things I’ve played around with, like APS theory and the log category, and I hope that with a better understanding of semisimple groups I can come back to these and put more life into them.

How much I’ve learned this quarter remains to be seen, but I’m definitely enjoying it a lot.

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