What Is Research?

July 24, 2006

Selecting a problem to work on

Filed under: Thinking and research — vipulnaik @ 8:12 am

Research (according to Wikipedia)is often described as an active, diligent, and systematic process of inquiry aimed at discovering, interpreting and revising facts. The “discovery” part of research has always fascinated me because of its connotations of “charting out new territory” and “exploring the unknown”. I want to know: how does one determine the right direction to proceed in order to discover new stuff?

Research activity was classified by WT Gowers into problem solving and theory building. The approach of researchers often varies between these two “ends” of the spectrum. Compulsive problem solvers, like Paul Erdos, feed themselves with an unending supply of problems, and churn out the solutions just as quickly. Erdos was legendary not only for his ability to quickly solve and move between problems, but also for his ability to identify, and set rewards for, the problems which required more effort. The apotheosis of the theory-building extreme is Grothendieck, who spent a concentrated thirteen years creating the “foundations” of algebraic geometry.

“Erdos style problem solving” has a more childlike appeal to it than the grave “Grothendieck style theory building”. Also, problem solving has a higher “discovery component” compared to theory building, which also involves interpreting and revising existing knowledge. For a person (like me) who’s just entering the research world, the problem solving route seems more attainable.

Hence the question: How does one select problems and go about solving them?

What is an open problem? It is a problem that hasn’t yet been solved. Open problems could be:

(i) Conjecture: A conjecture is an assertion which the “community” believes in, but which has been neither proved nor disproved. For instance, the Riemann hypothesis, the Poincare conjecture, and the Goldbach conjecture.

(ii) Gamble: A gamble is a yes/no open problem where the “community” does not believe in either a yes or a no.

(iii) Loosely formulated problem: This is an open problem that has not been formulated precisely. For instance, in classification problems, the meaning of the term “classification” is usually ambiguous. A loose problem formulation often indicates that the solution requires more of “theory building” than “problem solving”.

Should a researcher always pick open problems for research? An open problem is not the only option for researching on. A researcher can work instead on “interpreting” and “revising” existing knowledge by giving yet another proof of a known result or patching up an existing proof.

But an open problem carries with it the whiff of mystery, and solving it means expanding the body of knowledge and uncovering new territory. It is one thing to say “I found yet another proof of the Fundamental Theorem of Algebra” and another to say “I proved the Fundamental Theorem of Algebra”. It is one thing to say “I classified all non Abelian geometries” and another to say “I have recompiled the classification of all non Abelian geometries”.

So the next question: How should a researcher pick his/her open problem? Picking an open problem has two parts: picking the problem, and picking the stand on the problem. Picking the stand is particularly important. A researcher cannot just say “I am going to solve the Goldbach conjecture.” He/she says: “I am going to prove the Goldbach conjecture” or “I am going to disprove the Goldbach conjecture.” In fact, he/she has to go further and say “I believe in the Goldbach conjecture because of this-and-this and I hope and plan to prove it using such-and-such.”

A researcher needs to believe in what he/she is setting out to solve. I don’t think it pays to say “A lot of people want to prove the Riemann Hypothesis, so let me do that”, or worse, “A lot of people want to prove the Riemann Hypothesis, so let me disprove it and become really famous”. Belief, both in the goal and the approach, is crucial for the researcher to carve the path towards the goal.

When I say believe in the goal, I don’t mean that a researcher pick a vague topic of his/her own choice, work on it in isolation, and come back to publish it. Rather, I mean that the researcher picks a problem which is both important to the community and lovable to him/her.

But how is it that some people struggle to find a problem they can fall in love with, while others go around solving problems left, right and center? The question puzzles me a lot, and I hope that the comments and subsequent posts will throw some light.

Do check out how these views tie in with the You and Your Research page.

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 12, 2006

Looking back, and forward

Filed under: Uncategorized — vipulnaik @ 1:00 pm

The VSRP programme already feels like over. All I need to do now is deliver a short and sweet presentation on what I have learnt here (for which I am making slides using LaTeX). The presentation is titled “Module theoretic freeness over the invariant subring” and it is open to all.

I’ll be in this glorious place called TIFR for just two more days. So it is time for me to collect myself, figure out what I can squeeze from ehre in two more days, and then start packing my bags for the life ahead.

One regret I’ll have is missing out on interaction with the faculty here. Apart from with my guide (Professor Dipendra Prasad), I hardly talked to any of the faculty members. Why was that? TIFR has many of the country’s, and some of the world’s, best researchers. Why did I pass up this chance?

One thing is that I really don’t know how to approach a big shot mathematician who appears busy in his (or her) office. Do I just barge in and start blabbering? Obviously not. Do I research the mathematician’s tastes and read his/her papers and then decide on a line of approach? Or do I hang around in the canteen, place myself strategically next to the professor, and try to get involved in a conversation?

So what opportunities have I had? Who are the people with whom I had openings to interact? One of them was Professor M.S.Raghunathan who is one of India’s best mathematicians. He gave the introductory talk, and he also gave a lecture series on “Lie groups”. But I didn’t really feel synchronized with that Lecture Series. May be he seemed too high to me….

Professor Indranil Biswas gave a lecture series on differential geometry, which I enjoyed. But that’s not a topic on which I have lots of deep questions. And Dr. Raja Sridharan gave a few talks in the beginning on algebra, which had a few nice proofs, but I again didn’t find anything to follow him up on.

Another factor that I think pulls me down in these things is my inability to capitalize on “social lunches” and “social dinners”. For instance, I don’t go down for evening tea (because I don’t take it) but in fact, evening tea is an important occasion for the students to meet profs and drop in casual conversation.

DP (my abbreviation for my guide) had invited all the VSRP students for post dinner snacks on one day, and during that interaction, he encouraged us all to get friendly. Another TIFR faculty member, Dr. Riddhi Shah, also dropped in there and talked to us for some time.

Anyway, what is gone is gone. It is back to the same thing: every body is busy in his or her own work, to build contacts and guides requires an extra effort and initiative. Staying in one’s own world doesn’t always help.

My plan for now: go and talk to DP tomorrow. (I’ve already fixed that up with him). Ask him more questions relating to the other areas of work, relating to the other repercussions of the reading work that have been done. These include:

(i) What all that we discussed about Lie groups goes through for algebraic groups?
(ii) What is the concept of doing things over characteristic $p$?
(iii) Finite groups, what are the tools used in the study of them?

I also want to say a few words about my idea on “Sequences” and try to open the door for further communication via email.

Finally, I want to figure out whether it will be appropriate to ask for a letter recommendation from him at a later stage.

Will post more after tomorrow.

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.

July 8, 2006

A breakthrough

“And then he saw her. Those hours of waiting under the moonlight, that pacing up and down, those nervous glances at the watch, those frantic glances at the sky, all seemed to fade out as he saw her gliding across the lawns. And then he realized that she had always been there… if only he had looked that way!”

I had a chat with DP the next day again. I had already sent him the writeup and I presented my ideas. He had marked some corrections in my piece — mostly incorrect symbols here and there.

I had used a little piece of notation: “BIR” for “Base is a Retract”. I am all for defining all my notation but I had split my work across multiple files and I had just sent DP the main file. In case you’re curious, by “BIR” algebra I mean an algebra with a retraction (viz an idempotent endomorphism) to the base ring. Every connected graded algebra is naturally a BIR algebra.

DP asked me what this means and I explained. But once he realized it was just something I had cooked up for the purpose, he lost interest. But it was a reminder of an important lesson I have often ignored: avoid introducing too many terms of your own.

DP gave me a little more pep talk. He said that I have made good progress, but it is high time to actually get down to the proofs. I told him that I had started going through it, but hadn’t achieved mastery over the proofs. We chatted for some more time, and before leaving, DP told me: I am happy that you are at last happy… it is nice to know you have started enjoying it. I won’t be here for the week but I think you’ll manage now.”

DP then left, and my progress continued. But somehow, it started losing steam again. This went on for 4-5 days, even as I was blogging out my experiences. Then, I pulled myself up and said “Hey! I am here in TIFR to learn and do great stuff. I am here to do great stuff and I am going to do it. And that great stuff begins with doing the task I have been allotted with full commitment.”

And then I asked myself: “With all these cycles of highs and lows, what am I really achieving? have my lows actually been all that low? Have my highs actually been genuine?”

I realized that my progress and commitment had been there throughout. The difference was that there were times when I had a feeling and a desire to do the paper and at other times when I was just plodding my way through. And when I was just plodding my way through, I was more prone to distractions. But, repeat repeat: progress had been there throughout.

But now there was a week left, and I wanted to do a lot. And I knew it : I was going to do a lot. I had spent a lot of time absorbing the background material: now was the time to plunge into the proof.

The next day, I just managed to master one part of the paper in an hour’s time. Yes, this hour came after a lot of days of struggling, but I know that I could have brought the hour on earlier if I had really wanted to. And I started consolidating on that hour.

And I have since been hard at work, pushing my way through the paper. There have still been distractions but there are now no regrets in my mind about having taken the paper. The paper is part of me: it is part of my history, my experience, and it defines a part of my present.

Which brings me to a question I want to discuss further with you: why does research take so long. Why are (as I had put it earlier) the few periods of activity interspersed so thinly in long periods of inactivity? Is it really that necessary? Is incubation time so high, or is a lot of the gap redundant? Do we set a limitation for ourselves when we say: “I have three years to do my B.Sc., two years to do M.Sc. and five more years to do research, so let me relax. It anyway can’t be speeded up”?

Do we give up our attempts to learn, lose heart, and wander here and there too easily? Or is that wandering necessary? Of course, these things differ from person to person, and situation to situation. I look forward to your posts. On a somewhat related note, you want to have a look at Steve Pavlina’s webpage.

Do give your comments…

Progress slows down again…

Filed under: Uncategorized — vipulnaik @ 12:46 am

I kept pondering and trying to read up and learn along the directions DP had suggested. But the problem was: I couldn’t get the richness inside my head. And I didn’t have a reliable source from where I could get all the information that was needed for the paper without getting swamped with detail.

I located online Lectures on Invariant Theory and started reading them. But again, progress was slow. Most of the things they talked about were not very closely related to the paper of Kostant, and reading this one, I started getting sidetracked. So I once again started spending my time catching up with other activities, while making more half hearted attempts at getting the flavour of the paper.

As twelve days on the paper were coming to a close, I started getting panicky. Almost half the time was up, and I hadn’t mastered even a single important proof! And I hadn’t even understood the statements of half the theorems. And I hadn’t met my guide except that once!

So on the night of Tuesday the 27th, I sat down and resolved to make progress with the paper. Just read it, plod through the proofs, get it going somehow. And that’s when it struck me: go in for blogging it! May be as I blog down my frustrations, things will clarify themselves and I’ll feel motivated to work.

The next day, I decided to mail DP telling him honestly how little progress I had made and asking his advice. Here is a portion of the mail:

Could you tell me what areas of
algebraic geometry to read up and how they play a role in the paper?

I would also like to have some more general advice on how one goes about
reading papers, as well as more specific motivations regarding this
paper. I feel that my progress so far has been
disappointingly slow.

Could we meet some time today afternoon?

I would also like to present my progress on the paper but I’m not
fully prepared for it, so could we meet regarding that some time later
this week?

That same evening, DP called me and talked for some time with me. He asked me what specific doubts I had about the paper. I said that I’m not getting the hang of it. I asked him how exactly algebraic geometry comes into the picture. I tried to explain what I had understood of the paper so far.

I explained what little I had understood of the use of algebraic geometry, but said that it somehow didn’t quite explain it to me. DP looked at me for some time and said that he himself couldn’t formulate an answer to that.

He then said that if I wanted he could study the paper himself in detail. But I said that I would prefer doing the paper myself and presenting my ideas to him. He reiterated that once I’d start cracking the paper, things would start falling in place.

“Invariant theory is an interesting and relatively new area and it is not all that difficult”, he said.

I must do this paper, I told myself. And it is not as if I have been doing it. But so far, I have had a half hearted approach, with it being a kind of formality. Now, I should do it with greater devotion and with love. Even if it weren’t my first love. If I wanted to make progress, I would have to take responsibility for it, own up to my work.

The paper I finally managed to prepare is to be found here.

Looking forward to your comments… I’ll post DP’s in the next post.

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