What Is Research?

July 14, 2006

A final slide show

Filed under: Places and events,Regular updates,TIFR (VSRP) — vipulnaik @ 12:07 pm

Today, the VSRP programme at TIFR concluded. My project was to read and understand a paper on “Lie Group Representations of Polynomial Rings” and I have mastered the first section of the paper. My guide was Professor Dipendra Prasad, who was also the VSRP coordinator. For brevity, I shall refer to him as DP.

In the last two days (Thursday the 13th and Friday the 14th) of the programme, each VSRP student gave a Concluding Presentation from material learnt during the programme. The audience for the Concluding Presentations included other VSRP (Mathematics) students, Professor Dipendra Prasad, Professor M.S. Raghunathan, Dr. Riddhi Shah, and other interested faculty and research scholars. The venue was room AG77.

The lecture schedule was tight. Nine speakers had to be accommodated into limited time: two there hour slots minus tea breaks. So, all in all, each speaker could be allotted only 30-40 minutes. That is a very short time to present what was learnt in a whole month.

On Monday itself, I finalized my Concluding Presentation topic with DP: “Module theoretic freeness over the invariant subring”. I had already written a document on this and some related parts of Kostant’s paper which can be found here. Moreover, this was an “in itself” subtopic of the paper and I felt that I could do reasonable justice to it within the short time provided. The question was: how should I present?

A blackboard presentation seemed the obvious choice. AG77 has six blackboards, or, more precisely, a sextuple blackboard. (Check the P.S. for more). So blackboard space was no constraint. Experienced TIFR lecturers used the board admirably and with dramatic effect.

However, time for the talk was limited, and from my own experience, writing on the board is tedious and time consuming. During the recent Microsoft Research Summer School, I enjoyed attending computer based presentations, some using LaTeX (converted to PDF) and some using Microsoft Powerpoint. I wondered: why not try making a LaTeX presentation for my Concluding Presentation?

Some mistakes/problems that I wanted to guard myself against:

(i) Too much time writing stuff on the board: Writing and speaking at the same time becomes a strain.
(ii) Multiplexing the board use: The boards need to be used for stating results, putting down notation, giving proofs and a number of other things. There are two ways of using the boards. The naive way is to just use the board like reams of paper and fill one board after the other. The more sophisticated way is to reserve some boards for important stuff and use other boards for scribbling and side stuff. Unfortunately, the clever way requires planning and careful thought and a lot of experience, which I didn’t have.
I wanted to use the board cleverly, but that would be possible only if I limited and clearly specified my use of the board. Then I could decide what to keep, what to erase and so on. I didn’t want to get trapped multiplexing the board incorrectly.
(iii) Not concluding properly: The time constraint was an important factor and almost all presentations overshot the time limit.
So I wanted to make sure that I conclude on the right note rather than just peter off with nothing more to say. Thus, I decided to prepare some breakpoints in between.
(iv) Losing the audience right at the beginning: This is something that happens in many mathematical talks, and I wanted to avoid it. I wanted to let people decide whether or not to listen to my talk only after they at least got to know what my talk was about.
(v) Letting people think they can correct me whenever they want: In a short and highly time bound presentation, the most dangerous thing that can happen to a speaker is getting sidetracked by people pointing out errors, making comments and suggestions, and asking for more detailed explanations. I determined that I would strive to avoid this.

With these points in mind, I began preparing my slides. Though I had had a lot of experience with LaTeX, and I had tried my hand at writing slides in LaTeX as well, this was the first strongly mathematical slide show that I actually planned to present. I first made a preliminary version of the file and printed a paper version. This paper version began with the Prerequisites, the Nice to Knows, and the Goals. Each piece of matter was classified as Setup, Goal, Question, Proof Idea, Ingredient, Exploration and so on.

I then started finding out how many of the prerequisites the audience knew. Since many of them were not aware and not comfortable with some basic terminology, I thought I must spend time defining it. But when I did a test trial of things, I discovered that the test trial spent ten minutes on just developing prerequisites. So I decided to skip detailed explanations of the prerequisites.

After this “trial” I decided to add two footers to each slide: (i) Expected time, and (ii) Blackboard use. This was meant to be a guide both to me and to the audience. In case somebody in the audience asked a time consuming question, I would just indicate the expected time for the slide, give a quick answer, and move on.

I also made a mental plan of board use. I would use one board column to store important notation, another board column to store important terms defined, and another column to store results that I have stated but for which I have not provided complete proofs.

Thus prepared, I gave my talk. Here are the slides.

There were no interruptions (except one asking for the slide to be repositioned and one asking a trivial question about term ambiguity). So I just went on and on. 35 minutes into the show, DP asked me to start wrapping up, and I replied saying that I was on the penultimate slide. I finished within less than 45 minutes, overshooting my time limit by only 5 minutes.

DP commented that while my talk presented the main proof ideas, it did not give examples, although I myself had, in my discussions with him, viewed the examples in great detail. He further remarked that a thrust of Kostant’s paper was: how to work through specific examples?

I replied saying that my presentation was focussed on a small segment of the paper. I considered the proof ideas to be the relatively hard part and viewed this talk as a way to get an interested person over the hurdle and get straight into the paper. While preparing the slides, I had not considered working out example cases.

In retrospect, I realize that working out some special cases would have given the newcomers to the subject a better flavour of things. I’ll keep this in mind for my next presentation.

If you have comments on the slides, or on anything about presenting slides, do post them here.

P.S.:The “blackboard area” has three vertically separated parts. In each part, there are two sliding blackboards — the front and the back blackboard. Moving one of these up moves the other one down. (Check out a sliding blackboard arrangement to get an idea.)

July 13, 2006

A sorrow… and a determination

Filed under: TIFR (VSRP) — vipulnaik @ 5:15 pm

My “What Is Research?” blog often gets overwhelmed with a “What Is Me?” flavour and this is not surprising considering that I am largely inseparable from my work. So it is with this post, written far into the night, as the sun sets on my VSRP at TIFR.

In absolute terms I have nothing to complain about. A person whose greatest worries are purely related to academics or to “what to do now to further myself?” is one of the lucky few in this world who isn’t besotten with problems. I haven’t lived a life of late night movie shows, I haven’t gotten high ever, I haven’t been spending my precious time and energy falling in love, I have maintained a reasonable diet, a good exercise pattern, and a reasonable sleep schedule that only gets compromised in case of work and only on a temporary basis. I don’t have squabbling parents or drunken neighbours or any emotionally sapping family problems. I have a reasonably fit body, an open mind (or so I think), and an honest and forthright personality. I love myself a lot, I don’t indulge in undue modesty or undue vanity shows. As far as I know, I am liked and respected by my colleagues and others whom I interact with. I believe in doing the best I can for myself and for the world around me, and I am open to redefining that “best” as time progresses.

I was determined to do mathematical research from quite early on, though I didn’t know what it meant. I had the hunch that it would involve uncovering structures and patterns through a combination of creativity and rigourous logic. Rich patterns that resided within the mind. But what did a career in research entail? How did one prepare for it?

Preparing for the Olympiads was a natural first step for me, because I had heard that getting through the INMO guaranteed direct admission into Chennai Mathematical Institute which was one of the best places in the country for doing mathematics, and also because I had heard that Olympiad experiences shape a mathematician. So be it. I prepared for the Olympiads, multiplexing it with school and with so called IIT JEE preparation. I made it to the IMO team in 2003, and from then on, was all set for a life in mathematics. And I finally did join CMI for my B.Sc.

But having joined there, I have been largely clueless. What is research? What is
mathematical work? What kind of work must I or can I do to build my research potentialities?

I can list a few: reading, writing, learning, attending lectures, interacting with mathematicians, trying to reformulate ideas. All these, I have been doing. But, there’s a big thing I’ve missed out on.

One important component of success in any area, I believe, is knowing the when, where, and what of things. And this is something I have neglected, at least relative to my other capabilities. For instance, there have been many opportunities in CMI, in IMSc, and in TIFR, to interact with people who have “been there, done that”. There have been opportunities for organized summer schools that I may have missed out. Not that I wasted my summers… at least not this one, but I might have found other ways of utilizing my time that would have gone into a lasting record.

Currently, my focus should be on positioning myself for a life in mathematics at a premium institute where I can fulfill my dreams. Naturally, the procedure for applying to a good university must occupy my utmost attention. Unfortunately, this hasn’t been the case. Of course, there is still “lots of time” and probably will be till the last date. But there are also “lots of other things” and if I want to make a dent, I should put give application a top priority.

Which brings in a number of deep seated issues.

First: Do I think that neglecting my actual studies in order to focus on applying is a kind of bad thing to do? Such as cramming for an exam and getting through with fake means? May be. At least, the residual of those ideas still remain. Secondly, I feel that all the application work is clerical and procedural, not the kind of work that I associate with research. Thirdly, I just don’t have the energy at times to do it… and that’s what I’m trying to recover via this blog.

All the above concerns are stupid. I know that. Applying to a good place is what will further my utility to society the most. And devoting my clerical efforts to that application procedure indicates my level of commitment to things. I can see the light now. I remind myself that the vegetable seller does his job even if we considers it clerical, even it it seems peripheral to the giant strides of humanity. The mother who spends time cleaning her child’s urine knows that while the job she is doing may in the short term be pretty “menial”, in the long run it is shaping a new individual who will contribute to and reshape society. The same should be the case with my application procedure.

Research is not just about doing it… it is about being it. It is about being in the right environment. You can do research in a crowded basti but it is much easier to do it in an environment that lives and breathes the subject. And if I am committed to research, I should be committed to reaching such an environment.

Which is what I plan to focus on now.

The questions I’d like to raise on now: what is the balance between the actual doing research and the being at the right place part? Where should the trade offs lie? Looking forward to your comments.

July 10, 2006

A happy ending?

Filed under: Regular updates,TIFR (VSRP) — vipulnaik @ 8:38 am

Something has clicked of late. Ever since I wrote my last blog post, I’ve been on a high, and no, I didn’t drink. I am here on a mission and am doing it well. And next time, I’m going to do it better.

For those of you who haven’t read the earlier posts, a quick recap. I have just finished my second year of B.Sc. (Hons) in Mathematics (and C.S.) in the Chennai Mathematical Institute. I am really keen on pursuing research in mathematics or allied areas. Currently I am at the Visiting Students’ Research Programme of the School of Mathematics at the Tata Institute of Fundamental Research. The programme started on 15th June and is scheduled to end on 14th July.

My guide here is Professor Dipendra Prasad and I am studying a paper by Bertram Kostant titled Lie Group Representations of Polynomial Rings. The paper is eighty pages, and I honestly don’t think I’ll be able to complete all the eighty pages. Currently, I have understood the first 20-30 pages.

When I first came here, I thought: “I’ll just sit down, start reading, work through the proofs, and keep consolidating my ideas as I go along. Even if I do three pages a day, I’ll finish it in the allotted time.” This is a common trap of reasoning: dividing the “total volume” by the “number of slots” to determine “how much” to do in each slot.

But it doesn’t work that way. For stapling sheets, or delivering milk packets, may be. But for reading a paper, it doesn’t work. For one, there’ll be many days when no progress is made. On other days, what has been learnt previously needs to be consolidated. And most often, as it happened to me, it just doesn’t seem possible to continue reading and understanding the subject.

So the question: how can reading a paper be planned? I’m still wondering. But even as I figure that out for the future, I have in front of me the pressing task of decently wrapping up my tryst with Kostant’s paper on Lie groups.

I am now in the process of finalizing my documentation for the paper. Here’s where I stand roughly: Kostant’s paper discusses three important situations, and gives sufficient criteria for each. I have understood most parts of the proofs of the criteria for all, but there are important gaps. What confuses me most is the way the paper keeps shifting between the “algebraic geometry” approach, the “Lie theory” approach and the usual manipulations with groups and rings (that I’m most confused with). I am often unable to figure out what ingredients are going into a particular proof.

Next, I need to review all the work I have done, and have a short talk ready for presenting to DP (short for Professor Dipendra Prasad) by Thursday-Friday.

After that (which may not happen during the programme here) I need to go through the remainder of the paper which discusses how to figure out whether a given situation satisfies the criteria.

I’ve got to wrap up and prepare for my talk with DP. I’ll post links to my own notes on the papers in my next post.

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!!

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.

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