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

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.

May 8, 2007

A summer in Paris — the ENS

Filed under: ENS,Places and events,Regular updates — vipulnaik @ 1:48 pm
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As part of an exchange programme between Chennai Mathematical Institute and the Department of Mathematics at Ecole Normale Superieure, three of the people in my batch (Shreevatsa, Arul and I) are spending two months at the ENS. We are living at the Montrouge quarters of the ENS, and our academic headquarters (so to speak) are at the main ENS in Rue d’Ulm.

Neither CMI nor the ENS has placed any academic expectations on us. They have basically given us some facilities and have asked us to fend for ourselves, making use of these facilities. The general plan is that each of us study some topic(s) under the guidance of ENS faculty, and we may possibly be asked to present what we have learned at the end.

Prior to the ENS, I have studied/interacted with faculty for a long period of time, at my own college (Chennai Mathematical Institute), the Institute of Mathematical Sciences, and the Tata Institute of Fundamental Research. Each of these places was very different in terms of size and atmosphere. CMI is a rather small and informal place — it has almost nobody except students and faculty (that is, very little administrative staff), and its only departments are mathematics, computer science, and physics. All the offices have glass doors and open out to the grounds. The Institute of Mathematical Sciences is a relatively larger place, with an often irritating central air-conditioning, many more office rooms, and a much more closed look to it. Though the departments are the same, the sizes are much more. There are big roms with coffee-table discussions. There are a whole lot more administrators, and the place in general boasts of a much bigger size than CMI.

Tata Institute of Fundamental Research is truly monumental compared to CMI, with departments including Mathematics, Physics, Computer Science, Theoretical Physics, Chemistry etc. Apart from the large number of academic faculty, there are a whole lot of administrators. There are huge living quarters in addition to the main institute building. Overlooking the sea, TIFR is both open and closed — open in the sense that the rooms open out to the sea, closed in the sense that it’s a centrally air-conditioned building and one can shut the outside world and concentrate.

That said, all these places had some overall similarities: the way in which students and faculty members interacted, the kind of food, the way people organized themselves, was quite similar. The Ecole Normale Superieure is proving to be somewhat different.

Unlike CMI, IMSc, or TIFR, the ENS is located pretty close to the center of the city; not that this says much, because the center of Paris is not as crowded or congested as the center of an Indian city. However, it probably reflects the general trend in Paris to have universities everywhere, not just in far-away isolated corners. The ENS has departments in sciences as well as humanities and has a total of over a thousand students, including both students completing the last three years of their five-year diploma (the French equivalent of a B.Sc. cum M.Sc.) and research students.

The mathematics department itself has some 60-70 faculty members as well as many other visiting faculty from institutes like Orsay.

One of the striking features of mathematics at the ENS (at least to a person who’s studied in India) is that most of the mathematics here is done in French. In fact, almost all discussions amidst students and faculty members is in French, and courses and talks are mostly in French. Talks are in English only when the speakers come from other countries (which again may not necessarily be English-speaking). This often leads to some interesting language problems and issues. For instance, to publish in journals outside France, one must write in English, and to learn about cutting-edge work done outside France, one must read English. Thus, most of the older graduate students, as well as faculty members, speak fairly good English, and can lecture in and understand English.

I was not completely taken unawares by this because during the International Mathematical Olympiads in 2003 and 2004, I had seen people from different countries write the Olympiads in their own respective languages — the Hungarians wrote in Hungarian, the Chinese wrote in Chinese, the Japanese in Japanese and so on. The only countries which wrote the IMO in English were India, Sri Lanka, Trinidad and Tobago, US, UK, and some Arab and African countries — basically, countries which imported much of modern mathematics from outside.

Aside from the language, another thing that greatly impressed me about mathematics at the ENS (fr whatever little I have seen about it) was the great professionalism and care with which people talked while lecturing. This may in part be due to the system of French education, where great emphasis is placed on presentation skills and where students are grilled orally by instructors on a regular basis. I hoep to understand better how the French present stuff by attending some talks here at the ENS — if they are in French, that’ll also be an opportunity for me to try deciphering French in real time.

Another nice thing about the ENS is its library (or bibliotheque, as it is called in French — the word librarie is used for bookshop). The library is pretty huge, with a lot of books both in English and in French. It also has an interesting system of organization (which I have not yet cracked) and a lot of helpful librarians). The place is also maintained in a way that a lot of people can do a whole lot of serious study there — and the librarians are very helpful with locating stuff.

Now as to my academic programme.

Dr. Olivier Glass, the academic coordinator for the exchange programme, told Arul and me (the two who are interested in mathematics) to contact the faculty members David Madore and Olivier Schiffmann.

Dr. Schiffmann sent us a list of possible topics which we could study over the summer, which included Schubert calculus, removing singularities, quantum groups, representations of quantum groups,quivers and Hall algebras, and Khovanov invariants. All the topics were very interesting, so Arul and I met Dr. Schiffmann on Monday (7th) and he told us a little bit about each topic. I enjoyed all of them and for some time was in a dilemma as to which one to choose. After some thought, i decided to pick on Schubert calculus, because I had been studying stuff on related lines for some time and I thought this would be a natural extension of that stuff.

I was and am also keen on studying quantum groups and I shall probably be going over to these if I am able to reach a point of closure with Schubert varieties.

Will keep posting as I get more and more of an idea of the life at ENS.

March 28, 2007

A summer in France

Filed under: ENS,Places and events,Regular updates — vipulnaik @ 2:06 pm

To set the context for this blog post, I am among three students from Chennai Mathematical Institute who will be visiting Ecole Normale Superieure, Paris for an exchange programme. This exchange programme happens every year, with the top three students from the passing-out undergraduate batch going for the two summer months (May-June) to the ENS.

I came to know about this some time in the month of November (of course, I unofficially knew about it long ago). For some time, the upcoming visit has been filling me with a mix of hopeful anticipation and a sense of dread.

Among the more basic issues are the issues of food, living etc. it seems there may be some adjustments needed on that front, but on the whole, it should be manageable. Then there’s the fact that I am visiting Paris, which is supposed to be one of the best cities in a variety of ways (I don’t really know much about these things, but I’ve been told this so I am looking forward to seeing the place for myself).

On the academic front, I need to find myself a guide (I’m not sure about the need to part but I guess that to make my stay academically useful it’s best to work under a guide) and then to follow up on reading some stuff under that guide. From what I understand, I’ll have to give a presentation at the end of it.

The great thing is that from what I have fathered, the ENS is among the best research institutes for mathematics across the world, and the academic environment there is likely to be good. The mathematics department is much larger than that of CMI, and contains some big names. And I have interacted with some people from the ENS who have come to CMI and I’m definitely keen to meet more of them.

On the flip side, I have the memories of my one-month stay at TIFR, Mumbai where I went for the Visiting Students’ Research Programme. I had a nice time there (Navy Nagar, Mumbai is a nice place and TIFR, situated right on the seashore, is particularly nice) and moreover I got computer access and I used that to do all the things I usually do at home and in CMI and more. And there were some other people who had come to the VSRP with whom I occasionally used to have intense discussions. And I got to meet some fine professors.

But for the paper that I had been assigned to do (viz ”Lie Group Representations of Polynomial Rings”) opccupied very little of my time — in fact, there were days on end when I hardly even touched the paper. True, I did put up a few intense spells on it, but I wonder whether these few intense spells were all that the academic component of TIFR was.

There were also the illuminating sessions with my guide, Professor Dipendra Prasad, but unfortunately, because I was not making good enough progress on the paper, these sessions could not be too frequent. Had I worked more on the paper, and perhaps on some other related things, I may have been able to extract more.

I’m wondering whether the ENS, France will be something like that.

The further complication is that, out there at the ENS, I’ll be in a foreign place where I may not know that much about how to interact with the people and what their social conventions may be. While I don’t think that this will lead to any major social gaffes, it could definitely hamper my comfort level in approaching people and in seeking them out. Also, since I’ll be in a far-off country, the number of sources of amusement, reassurance and comfort (if things aren’t working out) will be fewer.

Here at TIFR, for instance, even when the academics was getting too bad, I could always pass my time on the computer, or talk to the other VSRP fellows, or go out in the streets and in general meet up with other people. In Paris, I’m not sure how easy it’ll be for me to do such things.

On the other hand, I do have the advantage of more maturity, and also, I know that, like TIFR, I can enjoy and have fun and do a bit of work — I don’t have to do a lot of work just because I am going there.

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