What Is Research?

June 21, 2007

The ENS — wrapping up

Filed under: ENS,Places and events,Regular updates — vipulnaik @ 9:05 am

I’m now reaching the fag end of my stay at the Ecole Normale Superieure. Yesterday I gave a one-hour presentation of the work I did at the ENS, and I have also completed preparing a write-up related to my talk, which is available here.

Looking back on my stay, I realize that I found it very enjoyable, despite all the apprehensions I had initially about it. My apprehensions were numerous, including what sort of food I will get, whether I will have computer access, how I will manage to communicate in a place where everybody speaks French, and there will be anything interesting or worthwhile to study or do at the ENS. Food, as it turned out, was a needless apprehension — I was able to cook all my meals and besides, the canteen food wasn’t bad. Computer access was not a problem at all. Regarding communication in French, i did pick up a little, and was able to read and understand the signs. But the design of Paris as a city allowed me to get away quite a bit without having to speak much. Paris is a city designed for self-help, unlike most Indian cities.

My academic apprehensions also turned out to be largely unfounded. First of all, I was able to spend most of the time just the way it was in CMI — working on my own, reading books and using the Internet, writing up and communicating with people via email. But the ENS gave me a few added advantages. First of all, they have a good library and they have JSTOR access and access to some other journal papers, which means that I can freely download papers relating to any subject/topic that I am studying. Secondly, there are a number of talks and seminars at the ENS, often in subjects that are different both in content and style from the ones I’ve attended in CMI. Some of them are in French, so that means an added challenge of understanding language.

The best part was that I got an excellent advisor, Professor Olivier Schiffmann. I’ve met him only four times so far (apart from the first time when he gave me a list of topics to study). But each time that we met, we talked for at least two hours, usually discussing a wide range of things.

In fact, I’d often go with a range of things to ask, some of which were doubts with steps in papers and books that I could not understand. But I would also pose some more open-ended questions to him, such as “What is the relation between all the things that are termed Hecke algebras?” or “What can we say about the analogue of Hecke algebras with respect to the parabolic subgroups?” or “What exactly is the relation between representations and sections of the line bundle?”

It was often in answer to these questions that Professor Schiffmann would tell me some loosely related stuff, and introduce me to new areas and connections I had not thought of. For instance, in response to my question of why so many different things are termed Hecke algebras, and whether there’s a unifying definition or notion for them. Professor Schiffmann explained that the original notion was probably that of Hecke operators in number theory, and that this related to the Hecke algebras we usually studied by means of the relation between number fields and function fields. This led to a lot of other interesting related ideas.

Another time, I asked Professor Schiffmann about the hecke algebras for parabolics, and he also mentioned that we can talk of different parabolics (other than the usual ones that preserve flagas) in the context of affine groups. he said that these often arise in physics.

My meetings with Professor Schiffmann thus helped me expand my vision of mathematics. It was a kind of expansion and elaboration that I would not have been able to achieve myself within such a short period of time. However, it’s also true that if I had not gone with so many questions, and with a sort of agenda in mind, then I would have been able to derive much less from meetings with Professor Schiffmann (probably, say, only half).

These have also reinforced a lesson that I have been learning repeatedly over the past few years, viz, it’s always upto oneself to find one’s path in life. People around can guide and advise, but the more you push for things, the more you get them. I used to wonder earlier about whether, once I start my doctoral research, I’ll be able to choose my path in life. I often thought between two extremes: doing my “own thing” (which I’ve always fancied) and “following a path set by others”.

But what I’ve learnt is that the real world is somewhere in between — it’s neither about doing one’s own thing nor about following a set path. rather, it’s about finding an “acceptable” path that one likes. In other words, I can’t go and tell somebody “I submit myself to you. Guide me, I’ll follow you” but I can’t say “I’ll do what I want and you don’t interfere”. It’s more of something like “yeah, here are a lots paths available and here is something I want to do. These are the resources I have at my disposal, and this is the goal that attracts me. How can I best use these resources to achieve the goal?”

Which is in some sense more difficult than either openly being different or blindly following, because it involves making a number of mild adjustments to get the maixmum (or at least a good amount of) mileage out of the things and resources around us. For instance, there may be only talks in a particular area of mathematics over a certain period of weeks. Or the advisors or people i get may be interested in discussing or helping me out only in certain areas. Or there may be other constraints. Now blindly following would just mean attending (or may be not attending) what courses are given, following whatever the advisor tells one to read, and so on. Carving one’s own path may mean deciding not to attend talks and courses outside one’s area of interest, and probably ignoring or neglecting (or procrastinating over) any work given by the advisor that is not in one’s area of interest.

But the thing with a research life is that while there’s a lot of pressure to do something, there’s usually very little pressure to conform to a particular thing. So if you don’t do the things that your immediate neighbourhood and facilities offer, then you end up doing nothing, and that’s what often used to happen with me (luckily for me, I haven’t yet entered research life, so nothing gained or lost yet). On the other hand, since there’s usually very little pressure to conform, advisors, guides and courses generally lose interest in people who are just blindly following.

So at the end of the day, it’s the student who chooses the direction, and directs the work. True, a lot of Ph.D. work is related to completing research work of others, and filling in gaps in others’ work, or working out in detail ideas of others. But even there, it is for the research student to choose and decide that the work and ideas originating from another person are important enough to take up and pursue to completion.

I hope that my experience at the ENS will stand me in good stead for later research life in mathematics, and also teach me the lesson not to be unduly apprehensive about visiting new countries and adjusting in new environments.

June 1, 2007

A skill set for research?

Filed under: Advice and information,Thinking and research — vipulnaik @ 9:15 pm

Looking back on my three years at CMI, and looking forward to five years of solid research in mathematics, I am currently trying to gauge: what is the skill set that I need to push myself through in mathematics, to see my ideas through (as I mentioned earlier), and to, in general, do well and survive?

Many of those things, I probably already have (though there’s always scope for improvement). For instance, I definitely have an ability to persevere with stuff, to concentrate for long hours. I also have a general propensity to document and organize ideas, to handle huge masses of information and ideas, and to come up with searching questions and search for answers to those.

Probably one of the things that I lack (not really lack, but could get more of) is the ability to do research alongside others, viz, to collaborate, to learn from senior people working in the area, to be able to work under another person’s guidance. I somehow feel (as I probably mentioned earlier in this blog) that I have too much of an independence trait, too much of a desire to do what I want, that may be it gets in the way of following what others say or advise.

Interestingly, I still do seek advice often; only I don’t usually implement it! I also often get started, half-heartedly, on a number of projects, and while I do tend to persevere on a few of them and see them to completion, I don’t usually close, or discard, the other projects.

Apart from being able to collaborate with people on an intimate basis, and take advice, i also need to get more on the social network of mathematicians. I think there’s quite a lot of sub-networks within mathematics students wherein the students share not just mathematics, but a lot of other things, and getting into these networks will help me feel part of a bigger community. Not having been able to do this much hasn’t really been my fault — on the other hand, I think I could have done better if I really wanted. It’s also something not unique to me.

I remember right from the time I was part of the Indian contingent to the International Mathematical Olympiad, that there were so many peopel from other nationalities whom I could have interacted with, but didn’t. It’s the same story out here, in France, four years from that time. Probably it has something to do with reverting to one’s familiar shell, doing the things one is typically used to (including both work and fun) rather than exploring what is new.

Another skill that I need to pick up is my presentation skill. As such, i am fairly good at giving mathematical talks, but presentation skills mean a lot more than just giving talks of mathematical content — it means presenting oneself to anybody in such a way that one can bring that person to one’s point of view! For instance, it may mean convincing a professor to back my research project, it may mean convincing somebody to sponsor me, it may mean a whole lot of other things!

Also I need to be a bit smarter about what the requirements at a place are, what it takes to convince the people of my sincerity as well as of my overall appropriateness. In CMI, I did suffer a bit initially because there were certain courses that I did not take seriously. At that time, I did not think that the knowledge I gain in them is of much use (which was probably true, but now I realize that coursework is an opportunity to gain a relationship with a teacher which could be useful later). I plan to be much smarter now. (In general, being smart about what counts and what doesn’t is an ingredient for success in any endeavour).

Research and funding — some questions to be answered later

Filed under: Places and events — vipulnaik @ 8:26 pm

Today, I attended a Psychiatry Seminar at College of France. This was a seminar meant for the public, with short 15-20 minute lectures by a number of eminent people in psychiatry research (most of them working with INSERM in various colleges and universities in Paris, but there were also two outsiders who have talks in English).

The talks were about research in psychiatry, mainly focussing on the following things: schizophrenia,bipolar disorder, and suicide. Issues like the cause of these (genes versus environment), the relation between psychiatry and neurological symptoms, the incidence of these in the population, and the effect on behaviour and emotional pattersn, were discussed. Though I didn’t really follow all of them (both due to language and subject gap) I definitely got a good general idea.

I noticed some fairly basic differences between the way psychiatry stuff was presented, and the way I have seen mathematics stuff usually presented. For one, almost every slide of the psychiatry talks had references to studies! In fact, every claim made was substantiated by the name of some study, and usually a lot of effort was made to establish the credibility of the study for important claims.

Mathematicians also enjoy cross-referring to one another, but in mathematics, it is not mandatory to refer to a past paper or publication whenever using results first mentioned in it. This is probably because mathematical arguments (at least, the simpler among them) can usually be explained and followed on the spot; it is hard to do the same for psychiatric arguments.

The problem with psychiatric arguments is that almost any statement about human behaviour can be justified, or made to appear convincing, to at least some people. While in mathematics, the problem may be that intuition is hard to get, it is probable that in subjects like psychiatry, one may have too much intuition — only it may not always square with reality! Further, personal experience and beliefs, while they may be a very reliable way for making personal decisions, cannot be quoted authoritately at other people. Which is why psychiatrists need to use established studies to justify, or establish, any statement.

Another thing I was struck with, initially, was the fact that a lot of the psychiatry research seemed to be about fairly pointless things, or at least, about things that didn’t seem to have much of a direct impact to actual psychiatric treatment. However, a little shake revealed that I had no right to say such things as a mathematician! Which led me to some general introspection about the research world.

The world of research and academia usually functions like this sort of closed system, that takes funds from outside regularly at the one end and supplies the other end with some tidbits of knowledge and information at the other end. But the system doesn’t just work in the way that it takes funds from outside and sells back knowledge or information — it’s not just like selling things in the market. It is more like keeping an entire system alive, throwing in money in it at times, getting rewards from it at times.

And the system of research and academia, like any other, develops its own tentacles, its own bureacucracy, its own hierarchy, its own conventions. For instance the world of mathematics research can be thought of as having tentacles in math departments across the universities of the world, special math-dedicated institutes, journals devoted to mathematics, special research groups, and others involved in doing mathematics. There are whole hierarchies of thought, whole resources devoted to mathematics. There are conventions of whta is good mathematics and what isn’t. In other words, the world of mathematics research is some huge ecosystem of its own — except that it needs its supply of money from outside, because it can’t directly make money, and also that often the results it produces shake the workd a few deacdes down the line.

In some other subjects, the relation with the outside universe may be more intimate, and hence the inputs as well as the output may be more closely correlated. For instance, when the French government sponsors a study on the causes of autism in children, they hope that the study will produce results that will help improve the quality of children’s lives. Or at any rate, they hope it may do so. On the other hand, when the government sponsors CMI people to do research in complex algebraic varieties, they aren’t hoping for any immediate gains to anybody, rather, they are just hoping that the general body of knowledge would have improved.

Which leads to the question: what if those who are keeping the ecosystem alive by pumping in the moeny, suddenly realize that they’re not getting their money’s worth out of it? Or what if they ask the research world to change its practices for greater apparent gains in results? Or to put it another way” how much is the research world, or research community, making sure that those who are pumping resources into it are kept satisfied with its performance, viz they get their results?

I think this is where the small differ from the big. When one asks big money, and promises big results, one is basically saying — “Okay, we’ll create a flourishing ecosystem if you pump stuff into us, and this is what you’ll get out of it!” On the other hand, if one asks small money, and zero interference, one is basically saying — “Okay, give us whatever spare money you have, and don’t ask us any questions”. The first is asking for an investment, the second is asking for alms or charity.

It is an interesting puzzle to me how mathematics research has been lasting for so long — is it simply on the chiarity basis, or do people view it as a worthwhile investment? My feeling is that mathematics research has grown too big, and too influential, to be simply cut down or removed. Thus, even if mathematicians do not produce any direct results of use, they are so closely tied to the other ecosystems (which are of very direct use) that it’s best to keep them alive!

For instance, work by mathematicians in number theory, algebraic geometry, and group theory, is used in areas of computer science like algorithms, complexity and cryptography. The relation isn’t just of one output being processed as another input, rather, it is the fact that the comuter scientist working in these areas often seeks to learn the mathematics, and actively engage in discussions with the mathematician, to progress on his or her work. Similarly a lot of work in differential and riemannian geometry, as well as much of Lie group theory, has profound relations with physics in the new era, and the physicists (thouh they useslightly differernt jargon) are always keen to understand more of the mathematics involved.

So basically, suppose somebody decides not to fund the mathematics department to do mathematics research. As such, there may be no immediate loss. However, the people in the physics department may suddenly find that there is nobody to teach courses to their physcis students in Lie groups and Riemannian geometry. the people in the computer science department may find themselves unable to get a good supply of new ideas in their areas.

In some sense, the security of mathematics as a research discipline lies, not so much in the direct utility of its results, but in the link with at least two disciplines: physics and computer science. Another important reason why people do not want to destroy mathematics departments, of course, is that people who graduate from mathematics departments often do well on a number of areas involving finance and statistics, and investment/actuaries! And these are sure big money!

For instance, a fair bit of CMI’s funding comes from companies which hope to get recruits into CMI after the Masters program in Computer Science. Actually ,the Masters program in Computer Science, while fairly good, is not the highlight of CMI’s programme, and definitely, most of the CMI faculty and resources do not go into it! But by having this one crucial likn with the outside world, CMI helds keep itself afloat, gets a certain amount of sponsorship.

Thus, in some sense, mathematicians are the cleverest, they get their funding from people, not as charity or investment, but rather as a kind of indirect investment, which means they are not accountable to anybody directly! The mathematician may be imagined as the person who says — “You need me to be around, but I can’t be around unless you give me food to eat, and unless you give me a car to drive in, and unless you give me a house to live in. I don’t need these for your work, but these are my conditions.”

Unfortunately, this, though definitely a better attractor than charity, stil doesn’t get huge funds (which is true, after all, for the mathematics departments). The best funds will come in if the mathematician makes a case that the particular research being done is of use, rather than the give me the money to keep me happy line. I am not sure which mathematical projects qualify for loans and grants because of their particular importance. I am sure there are a few, but it certainly doesn’t seem to be something that most mathematics projects can boast of!

In coming blog posts, I will look at how unuversities, and funders, decide to support research, and how this differs across various subjects, with particular emphasis, of course, to mathematics.

Ideas — and what it takes to see them through

Filed under: Thinking and research — vipulnaik @ 7:35 pm
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In an earlier post on this blog, I had outlined a so-called APS theory, which had roughly the following highlight: we take a sequence of sets and define maps from the set corresponding to m times the set corresponding to n, to the set corresponding to m + n. Sometime in the second week of May, I had discussions with a student Olivier Dudas, and with my advisor Olivier Schiffmann, where I learnt some interesting things about the general linear groups and orthogonal groups. These tidbits that I picked up led me to come back to the APS theory. I looked at it again and found that many of the ideas in this theory could be developed further by me now that I know a bit better about things like Dynkin diagrams, root systems, Chevalley groups, Lie algebras, and the peculiar relations between the general linear group and orthogonal group.

Towards this end, I focussed on exploring more closely the rough sequences-theoretic notion of algebraic extension that I had developed earlier. Roughly the idea is that just like we say that a field is an algebraic extension of another field, in the same way, we should be able to talk of algebraic extensions of APSes, that generalizes that of fields. In essence:

  • If L is an algebraic extension of K of degree d, then the APS of general linear groups over L is an algebraic extension of degree d over the APS of general linear groups over K.
  • The general linear APS over any field, is an extension of degree 2 over the orthogonal APS over the same field.
  • The unitary APS over the complex numbers is an extension of degree 2 over the orthogonal APS over the real numbers.
  • If we have an algebraic extension of APSes of algebraic groups, then this also gives an extension at the level of their Weyl groups. Thus, the fact that the general linear group is an extension of degree 2 over the orthogonal group, translates to the fact that the symmetric group APS is an extension of degree 2 over the signed permutation APS.

Equipeed with this and many other related areas, I went and talked to Professor Schiffmann. I also asked him whether this APS-theoretic study can be extended to other related structures and constructs such as the associated Hecke algebra. Professor Schiffmann was quite enthusiastic and interested, and suggested that I look at the correlation between this and invariant theory.

In particular, he focussed on the part about how the general linear APS over complex numbers was a degree 2 extension over the general linear group over the real numbers, and suggested that this could have to do with the theory of invariant polynomials. he even gave me some concrete links. Unfortunately I haven’t been working on the theory, or on invariant theory, much since I last met him!

However, there were a few other things about the meeting with Professor Schiffmann that impressed me.

One was the fact that even though he was quite enthusiastic, he was very clear that “my” theory was my responsibility and my work and it was upto me to decide whether to spend (waste) my time with this theory! That is, it was up to me to prove the worth of the theory, the world was not going to take it up and say — “this is interesting, let’s develop it further”. This was clear from the way he always emphasized “You could look at this further in relation to your theory”.

Also, his emphasis was that if I want to give credibility to my theory, then I should forge strong relations between that and existing important results and paradigms in mathematics. This is something that I have felt myself, but his repeated emphasis on whatever interesting results and framework are already known, helped reinforce the fact that to be accepted into a body of knowledge, a theory must first integrate itself completely into that body of knowledge!

The above things may seem obvious in hindsight. On the other hand, there was a time when I was very young, when I had dreams of doing some funny thing like writing great stuff that everybody would want to read! Basically the idea that whatever I said, people would puzzle over and try earnestly to interpret and follow.

But I have realized that such things don’t happen. Life (and even research life) is not just about doing great things, it is also about advertising those great things, it is about putting those great things in a framework that others can follow, that others find easy to accept. In other words, it is about conforming to a system into which one wants to be accepted.

Tihs learning has come to me, in its true form, very gradually — and I am probably still learning it. In my first year at CMI, I thought — okay, if I have a great idea, then that’s it! But then I realized that a lot of effort goes into packaging an idea, into conveying it to the right people, into making it ingnite into something that is actually useful. I also realized that even a bad idea can be implemented much better and thus be of much greater use, than a good idea which one doesn’t bother to implement and work hard at. In fact, much of research, from what I gather, is fairly average ideas that people have bothered to implement to the point of completion. The great ideas may be sources of inspiration — which few but the best can implement!

This brings one to the interesting notion of seeing an idea through. This is the commitment to having faith in a particular idea (which may be a goal, a conjecture, a method, whatever) and doing whatever it takes to make that idea reach a stage where it is accepted by a larger community as being useful. There are, of course, lots of problems with this. Firstly, how does one judge beforehand whether an idea is worth seeing through? Secondly, why is it important to see an idea through?

The first question is important and interesting because most of us don’t have ideas, and even those who do have, rarely want to see them through (I do have ideas, but at this stage, am not sure whether I want to see them through). Rather, as doctoral students, we often may ask our advisors for the ideas and problems, which we then try to see through. Advisors in general may be better able to tell us what we is (a) more likely for us to be able to see through (b) more useful to the community if we see through.

However, I feel, and still believe, that if a person is able to come up himself or herself with something that he or she can see through to the very completion and which in itself has some impact (however small) on the way a community thinks, then that achievement counts for a lot more than following an advisor’s direction on picking what one wants to see through. Of course, it also has a lot of inherent dangers, since how can an inexperienced student know what is important, and how that can be attained? But that really is the kind of thing that I personally want to do — find something that I am truly passionate about, something I have discovered myself, and see it through till the very end.

Of course, I am not sure that the things I have come up with right now (or am toying with right now) are anywhere near the kind of things I can see through (though I think APS theory holds a good chance, but let’s not be too premature’ I have a lot to see and understand). I am also not sure about whether the people I get as doctoral advisors will have some agenda for me; on the whole, I’d rather get people who have some broad agenda, than people who just let me do what I please! But what I hope for is to find a setup wherein what I personally want to see through matches with the agenda that is set for me (upto a little adjustment on my side).

But I do feel I have what it takes to see things through. It is not clear where this confidence arises from. Probably it comes from the fact that generally, whatever I decide upon, I get done, even if not exactly in the way I had planned! Also, my efforts with the Extensible Automorphisms Problem, with my Group properties wiki (group theory wiki), and whatever I have done towards APS theory, have greatly incresed my confidence and my ability with processing and organizing information and solving particular problems.

What I want to explore right now, at the ENS, and in general, is: what is the skill set that I need to develop to be able to see stuff through? To be able to persevere at stuff, build stuff where it didn’t exist, to be able to convert an idea into something that makes an impact on a community? I will explore answers to these questions in the coming blog posts.

The ENS — more of it

Filed under: ENS,Places and events — vipulnaik @ 7:08 pm

In my last post on the Ecole Normale Superieure, I had mentioned that I am at its Department of Mathematics, for a two-month exchange programme with my undergraduate institution (Chennai Mathematical Institute). I have now been here for two weeks, and while I haven’t still explored the whole of the ENS, I have attended the seminars of some of the people here, and there were a few interesting things I noticed about the way talks are given at the ENS.

The first thing I noticed was that in the 3-4 talks I attended, the speaker gave the talk like a series of points. Basically something like Point number 1: (some part of the talk). Then Point Number 2: (another part of the talk). And so on. The talks didn’t seem to have an introduction, conclusion etc. in the conventional sense. Moreover, each point was focussed on examples.

Another thing I noticed about the way the French speakers talked, was that they in general were a lot more expressive than those people whom I have seen talking in India. Though my lack of knowledge of French impaired me from understanding anything but the basic outline of the talk, I could figure out that the speakers were using quite a bit of idiomatic language.

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