What Is Research?

June 30, 2006

The mental block

Filed under: Regular updates,Thinking and research,TIFR (VSRP) — vipulnaik @ 4:09 pm

My contrived success with the first paragraph of Kostant notwithstanding, I didn’t consider myself very successful with the paper. I wanted to get into the meat of things, and I hadn’t managed.

And for me, there has always been a huge list of other “to do” things. So when progress with the paper was lax, I started catching up with those other “to do”s, such as, updating my webpage, going out and meeting a friend, contributing meaningful articles to Wikipedia the free Encyclopaedia, and making myself useful to the community. Of course, with my lack of progress on the paper always at the back of my mind, gnawing at me. But guilt is something I have learnt to block out very effectively and the day swere so action packed that I hardly noticed how little time I was spending on the actual paper.

I wouldn’t say, however, that the time was a complete waste. You know how a tight can is opened, right? You try to lift it from one side, then loosen from the other and so on, till a critical amount is off, and then you just yank it. Now one way of doing it is to keep hard at lifting it and finish the whole activity in a single shot. The other is to try once, relax for some time, try again, and so on. Not so effective, but some work gets done anyhow.

One of the nice habits I developed from my early days in CMI was to keep documenting my observations. Actually the habit goes back to earlier when I used to prepare notes for the topics covered in school, though my notes at that time were understandable only by me. After joining CMI, I learnt how to write math stuff using the documentation tool LaTeX which you can download from here. This made it easy for me to typeset mathematical symbols and view a neat and clean version of my own creations and so I started using the computer more and more to document my knowledge.

Thus, even though I wasn’t relaly getting neck deep into Kostant’s paper, I kept recording my observations, and what I felt were the background motivations. My own ideas were probably at odds with Kostant’s, but at least it gave me the feeling that I was in control and doing something. Have a look at my initial findings right here. Sorry if that sounds like a muddle, that was what my mind said to start with! And also sorry if you’re wondering what Kostant’s paper really was, it isn’t legal for me to put it up online, so you’d better try getting it from a library.

Armed with this paper and a general feeling of inadequacy and uncertainty, I went and met DP.

Since I’d already sent DP my all too brief writeup in advance, I wondered: “Should I present what little I’ve written or ask him to explain some things to me?” I realized that although I had had a number of questions, I could not formulate them. Talk of getting tongue tied!

DP had already printed out my paper and marked errors. He seemed to have read the paper more carefully than I myself had. He had marked a number of errors. There was one thing he said he didn’t understand: the “Galois correspondence” that I had written about in my paper. I explained it to him.

I then explained to DP the procedure for computing the invariant subring of an algebra of functions from a space to a field, which brought algebraic geometry naturally into the picture. I used it to show DP that the invariant subring under the orthogonal group is the subring generated by the sum of squares polynomial.

I then confessed that I haven’t really made great progress. DP smiled at me and said that this result on the invariant subring under the orthogonal polynomials was an important (though easy) result and my coming up with the statement as well as the proof indicated that I had got the generic motivations correct. He then told me that invariant theory is an interesting subject and does not have too many prerequisites. Finally, he commended my habit of writing down all I was learning and told me to keep sending him updates.

The mathematics we discussed was very little. The main content of his advice was: “The cases yo uahve analyzed are the simple ones. The more interesting thing happens when the Lie group acts, not on the vector space, but on the Lie algebra by its adjoint action. There, we have a large nubmer of orbits of different shapes and sizes and understanding the orbits is a tough task. For instance, under the action of $GL_n(k)$, $k^n$ has only two orbits, but the Lie algebra which is $M(n)$, has a large number of orbits viz conjugacy classes.”

But it was inspiring and made me feel like working more on the subject. I realized the directions in which I need to explore. Discussions and guidance does help!!

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June 29, 2006

The eighty page paper — begins

Filed under: Regular updates,Thinking and research,TIFR (VSRP) — vipulnaik @ 4:01 pm

The paper “Lie Group Representations on Polynomial Rings” (sorry, you can’t read the paper unless you have JSTOR access, and it’s illegal for me to put it up online) is eighty pages long. It was penned by Kostant in 1963, so it is about 43 years old. So how do I begin with it?

Every daunting task should be handled by breaking it into submodules, so I decided I’ll just concentrate on getting the gist of the kind of issues that the author is trying to address. I basically decided to begin from the beginning — Page 1. (Page 1 is publicly accessible, so you can read it yourself at the link). Talk of symbols watering in front of my eyes!

One of the things that has irritated me in the books I have read nad the lectures I have attended is an excessive use of symbols like letters to denote abstract concepts. Symbols are indeed indispensable because without them, abstract algebra would have been impossible. However, I think symbols should remain what they are: symbols, and not synonyms for the concepts they connote.

What sends me up is statements like: In our talk, R will always denote a commutative ring with identity. Or worse still, people assuming that the letter N, wherever it pops up, means a normal subgroup, and not even bothering to say so. I feel this has its own dangers as we lose out on the statement being made at the conceptual level.

And Kostant’s paper was full of such statements…

This brings me to the old issue of mathematicians being accused of deliberately trying to obscure their work in technicality. Richard Hamming, in his classical talk, says that giving an accessible talk is difficult because it forces the mathematician to step back and critically examine how his or her work fits into a larger perspective. Staying immersed in one’s own comfort zones, however murky they may be, seems so much easier! We mathematicians use jargon as a means of protection (largely imaginary) against a world where we feel we don’t belong.

May be that’s taking it a little too far, … and any way, I didn’t know why Kostant had chosen his conventions the way he did. It probably had more advantages than disadvantages and I needed to first understand what he was saying.

So a look at the first para.

Let G be a group of linear transformations on a finite dimensional real or complex vector spacve X. Assume X is completely reducible as a G module. Let S be the ring of all complex-valued polynomials on X, regarded as a G module in the obvious way and let J be the subring of S comprising the G invariant polynomials on X.

Dense, it seemed to me. Which brings me to another question. What do we have against dense material which makes it more difficult than light material which may be much longer? I think it is the fact that we are not used to stopping at the end of each sentence and evaluating what it means. We want to use the pipeline of our minds: read the next sentence as the back of the mind evaluates the previous one. This pipeline is effective only if the effort required to understand each sentence is minimal.

But I’d been sitting in front of this page for quite a long time, so I decided: “might as well seriously try making sense of it. I probably do know enough for that.”

Read again: sentence 1 “Let G be a …” absolutely clear. I know a lot about subgroups of the general linear group and I’ve studied the representation theory of finite groups. So nothing new in that.

sentence 2: “Assume X is…” A mundanity. Forget it, I know what it means, but I don’t have thecontext to understand its significance.

sentence 3: “Let S …” Now I did remember that when a group acts on a vector space, it does also act on the polynomial ring. In fact, I remembered reading somewhere about the invariant subring under the symmetric group being the subring generated by the elementary symmetric functions, which is a formulation of the Fundamental Theorem for elementary symmetric functions.

So I thought: “This is stuff I probably know and can make sense of. But how do I keep track of and remember the symbols? What is the author actually trying to do and compute? May be I can try working and computing the things myself. Reading the paper passively seems a very dull activity and may be if I work things out, I’ll better appreciate what Kostant has done.”

Which raises many interesting questions abotu how we go about reading stuff. What should we do and what do we end up doing?
(i) Stare at it: Is it completely effective or does it set the stage for back of the mind processing?
(ii) Skim through it: If we’re not following anything, is it still worth skimming through it?
(iii) Decipher it sentence by sentence: How much time does that take? Do we lose sight of the big picture in that process?
(iv) Get a general idea and then explore through other means: That’s what I’m usually very comfortable with. But is it always appropriate? Isn’t there a chance of my going too far astray?

Looknig forward to your comments…

June 28, 2006

The TIFR experience (begins)

Filed under: Places and events,Regular updates,TIFR (VSRP) — vipulnaik @ 4:26 pm

I’ve been two weeks at the Tata Institute of Fundamental Research. This is as part of a month long Visiting Students’ Research Programme. The programme has been frustrating in a number of ways… but then, I think the good parts are yet to come.

Before coming to TIFR, I wondered why people supposed to be doing research end up doing nothing most of the days. Well, now, I know that even I am susceptible to that, thoguh I think my nothings are more worthwhile than theirs (compare contributing articles to Wikipedia, writing pages on Olympiads, blogging out my experiences, and other noble changes to the world, as opposed to spending time playing computer games). But then, that’s just opinion. The fact of the matter is that when confronted with the ocean, you prefer not to drink. And the same holds (or shall we say held?) for me.

So that’s enough of philosophizing, now for a detailed description of what happened to me. I’ll just document the beginning in this post. Stay tuned for more!

About a week before going for the VSRP, I got the sudden feeling that I need to get prepared for this thing. I mean, TIFR is the best research institute in India for mathematics, and it is one of the places I am considering for doing my Integrated Ph.D. after finishing my B.Sc. So I naturally wanted to squeeze the maximum I could from the camp.

At the time of my selection, I had sent in a list of topics I was interested in. But I hadn’t got to hear anything from VSRP about what work I would be assigned. So I decided to send this list again, this time directly to the person mentioned to the academic coordinator, Professor Dipendra Prasad.

My primary interest area within mathematics is group theory (more on that later). But I knew that in most places, pure group theory as a subject in itself doesn’t get all that attention, so I played safe by putting in many other topics I was also interested in. The top two points were:

(i) Group Theory and Representation Theory
(ii) Commutative algebra and algebraic geometry

Professor Dipendra Prasad (whom I’ll just call DP for brevity, without intending any lack of respect) replied promptly suggesting the paper on “Lie Group Representations of Polynomial Rings”. Talk of shoving in group theory, representation theory, commutative algebra and algebraic geometry all in one! The topic didn’t exactly set me on fire, but I decided that I might as well go for it.

Still zestful to prepare, I googled for the paper. Alas, no JSTOR access and no permission to search MathSciNet for me at home! So all there was to do was to wait to go to TIFR to read the paper.

On 15th June, I arrived at TIFR, all ready to begin my Visiting Students’ Research Programme. The official intro was at 2:30 p.m. but I wanted to get started as early as possible. So, along with a couple of other VSRP students, I dropped in to meet DP.

DP made all three of us sit in chairs (in TIFR, all offices have one cozy chair for the person and 2-3 hard chairs for others).

He began with the question, “Do you want to do something easy or something difficult?” After trying to play it safe, we concluded, along with him, that we might as well do something difficult and something different.

He then took a miniature interview of each of us.

He began by asking me my name, college etc. and wrote it on a piece of paper. Then his brow furrowed and he asked me if I was the person who had mailed him some time ago. I told him I had tried to locate the paper online but wasn’t allowed access. He said he’ll locate the paper and give it to me today, and I should get started on reading it.

“It is a bit long paper… but it will be good to read”. He fixed me with a penetrating stare and asked me if it was okay. I wasn’t sure but wanted to go ahead and try, so I said yes.

Interesting and exciting… well, somehow, I wasn’t too enthusiastic. But I thought I’ll give it my best shot.

Actually, there are plenty of interesting questions that I’d like to explore:

How are research topics chosen? Who decides what a student, fresh into “research”, studies? The student or the guide? Does the guide also need to take into account his or her own limitations?
What happens to students whose areas of interest are not catered to by the guide or by the institute? Should they pursue their interest or do something where they can get maximum help?

DP is a fairly versatile all rounder with at least a basic knowledge of all subjects, and he was willing to take practically all the students. But what if it had been somebody else who had only a small compass of knowledge?

In Hamming’s talk on research he says that we should work on the small problems in the important areas. Two mistakes he tells us to guard against are: (i) Working in unimportant areas and (ii) Working on massive problems

His advice seems targeted as the student studying for or after his doctorate. What about undergraduate students who are yet to garner experience in all areas?

Keep reading as I unfold my opinions on these issues. And do respond with your own comments and posts.

The question has been bugging me since…

Filed under: Uncategorized — vipulnaik @ 4:11 pm

Fun and inactivity accompany us wherever we go, but it’s the serious work we do in between that counts”

I’ve just finished two years of B.Sc.(Hons), Mathematics and Computer Science at the Chennai Mathematical Institute (CMI) in Chennai, India. In these two years, I have interacted with many researchers, some in my college, and some at the summer camps I’ve attended: Institute of Mathematical Sciences (IMSc), Microsoft Research Summer School on Algorithms, Complexity and Cryptography at Indian Institute of Science (IISc), and (currently attending) the Visiting Students’ Research Programme (VSRP) at the Tata Institute of Fundamental Research (TIFR) School of mathematics.

Yes, and I plan to do research. So far I’ve only got a few whiffs but it’s addictive… or so I’d like to think. But it seems frustrating. The ideal picture one has of a researcher is a person who’s intensely thinking about his or her problems all the time, taking them to the coffee table, pondering them over in the bus ride, pacing up and down, Googling, surfing the net, drawing doodles on the board, and so on. Indeed, that’s research life… but these intense moments of activity seem to be embedded in large periods of inactivity where more mundane considerations rule the roost.

So how does a person become a top class researcher? Is it intelligence? Hard work? Creativity? Sheer determination? A conducive environment? How do researchers pick on open problems, how dothey make serious efforts to solve these open problems? These are issues I plan to discuss in these blogs, and my discussions will begin from where I am: the personal experiences I have had while trying to resolve these questions during my summer camps.

The posts on this blog are organized as follows. Each post begins with either some incident or event relating to my mathematical life, or with the story of one of my own mathematical ideas and how I developed and perhaps discarded it. Towards the end of the post, I raise a number of questions of the kind where both questions and answers are thrown up by my blog.

But before you hear me, you might like to have a look at the talk by Richard Hamming the expert on You and Your Research.

P.S.: If you really want to know more about me, visit my Home page.

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