What Is Research?

March 8, 2008

Tryst with functional analysis

It’s the end of the ninth of the eleven-week winter quarter, and the next two weeks are probably going to be fairly hectic: we have examinations/final homeworks to submit in all subjects, and I’m guessing that from tomorrow onwards, work on these will begin full-force. So I’m taking a little time off right now to describe my tryst with functional analysis so far.

During the first 1-2 weeks of the functional analysis course taught by Professor Ryzhik, I was enjoying the material, more or less keeping pace with the material, and also reading ahead some topics that I thought he might cover. However, from around the third week onwards, the nature of topics being covered in the course changed somewhat and I started getting out of sync with the material. Then came an assignment with problems that I had no idea of how to solve. Eventually, solutions to these problems were found in an expository paper by David (one of my batchmates) and the first years worked out the details of the solution on the chalkboard.

At the time, I was feeling tired, so I didn’t try to keep pace with and understand all the details of these solutions. I wrote down a reasonable bit of them to muster a decent score on the assignment but I didn’t internalize the problem statements (I did have some ideas about the problems but not from the angle that Prof. Ryzhik was targeting).

So, in the next week’s problem set, I wasn’t able to solve any of the problems. This wasn’t because the problems were individually hard (though some of them were) but because even the easy problems needed a kind of tuning in that I hadn’t done/ I learned of the solutions from others and understood enough of them to submit my assignment, but they hadn’t sunk in. At the same time, I was handling a number of other things and I didn’t have a clear idea of how to proceed with studying analysis.

Some time during this uncomfortable period with the subject, I remembered that the previous quarter, I had overcome my discomfort with noncommutative algebra by writing Flavour of noncommutative algebra part 1 and Flavour of noncommutative algebra part 2. Noncommutative algebra differed from functional analysis: in the former, I was reasonably good at solving individual problems but just hadn’t had the time to look back and get the bigger picture. In functional analysis, I didn’t start off with a good problem-solving ability or an understanding of the bigger picture.

Nonetheless, I knew that trying to prepare a write-up on the subject was probably the best way of utilizing my energies and probably a way that would also be useful to other students, which could partly be a way of contributing back, considering that I hadn’t solved any of the recent assignment problems. Moreover, it was something I knew I’d enjoy doing and I hoped to learn a lot from. So I got started. The first attempt at preparing notes was just aroudn the corner from the mid-term. I got a lot of help from Rita Jimenez Rolland (one of my batchmates) who explained various parts of the course to me as I typed them in. (Here’s the write-up).

However, after the examination (where I didn’t do too well — notes are more useful if not prepared at the last minute) and as I learned more and more of the subject, I felt that it’s good to restart the notes-making process. I brainstormed myself about what kind of write-up would be most useful. Instead of just trying to cover whatever has been done in the course, I tried to look at the problems from a more basic angle, like: what are the fundamental objects here? What are the things we’re fundamentally interested in? I also brainstormed Mike Miller, who provided some more useful suggestions, and I got started with the write-up.

Preparing the analysis write-up hasn’t been plain sailing. The problem isn’t so much lack of time, as it is lack of richness of engagement. When I’m working on my group theory wiki or writing this blog entry, or doing something where I have a very rich and vivid idea of what’s going on, every part of my mind is engaged. There isn’t scope for distraction or going lax, because I’m engaging myself completely. However, when writing functional analysis notes, I faced the problem of my own ignorance and lack of depth and ideas in the subject. So, when I got stuck at something, I didn’t have enough alternate routes to keep myself engaged with the subject. The result? I kept distracting myself by checking email, catching up with other stuff, and what-not.

The contrast was most striking some time about a week ago. Through one hour of interrupted and not-very-focussed work on the functional analysis notes, I was getting somewhat frustrated. On a whim, I decided to switch to working on the group theory wiki. I did that, and was surprised to observe that for the next one hour, I didn’t check my email even once.

The complete concentration on the subject isn’t merely explained by the fact that I like group theory more, or am better at it. It is more the fact that I can see a bigger picture. Even if I’m concentrating on a couple of trees in the forest of group theory, I can see a much larger part of the forest. But when working on a couple of trees in functional analysis, all I can see is those and a bunch of other trees. So distractions find their way more easily.

I consider this illustrative because we often think of concentration as a kind of tool of willpower. True, the exertion of willpower is necessary to concentrate at some times (e.g. to pull myself back from the group theory wiki and back to functional analysis). But more fundamentally, I think it’s the intrinsic ability to see something as very rich and beautiful and to keep oneself completely engaged, that matters. Do determination and hardwork play a role? Yes, they do, but they do so because they help build that internal richness. Which explains why I love writing so much: in a number of areas, writing allows me the most to explore the inner richness. And I think this is a factor in explaining why, although many different people work hard, there are not so many who, at the end of their hardwork, find the work enjoyable. That’s because most of us use a very small part of the tremendous hardwork that we put in, into creating an internal richness that can engage us better.

What about functional analysis and me? Do I see the richness in functional analysis yet? Not to the level that’d help me cope very effectively with the course, but yes, I do feel a lot better about the subject. And I think the new notes on function spaces, even though they may seem amateurish right now, do indicate some of the insight and richness that I have gathered over the past few weeks. Let’s hope I can augment these notes in the coming days to a level that really gets me prepared for the examination!

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