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.