# What Is Research?

## July 7, 2009

### Million words on the first-year: Mocumentary of first-year life at Chicago

Filed under: Chicago,Places and events — vipulnaik @ 10:10 pm

It is said that a picture is worth a thousand words. A six-minute video is probably a thousand pictures.

There is a tradition in the mathematics department of the University of Chicago whereby, at the end of each academic year, the second-year class organizes an evening of skits called “Beer Skits” (“beer” because the skits are accompanied by huge servings of beer). As part of the tradition, we decided this year to create a mocumentary (mock documentary) of life in the first year at Chicago. I had the original concept and fleshed out the main scenes, but a lot of the editing work before, during, and after the shooting was done by many of my batchmates, who all added their own insights and removed some of my original ones that would have been too abstruse.

Our six-minute video is a little funny, but it is also quite realistic, as those who have gone through a similar experience can attest. Okay, agreed, some of the scenes towards the end stretch the boundaries of realism, but almost everything we have is a slight adaptation of something that actually happened during our first year.

Below is an embedded Youtube video (you can make it full-screen for better viewing).

## March 29, 2008

Filed under: Advice and information,Places and events — vipulnaik @ 2:37 pm

Yesterday, I went and attended a “panel” held at the Chemistry Department about science students who’ve gone on to unconventional careers related to science, outside both academia and industry. A few weeks ago, there was a panel discussion between us grad students at the math department and four post-doctoral students. And I’ve been watching some panel discussions on videos on the web. Based on this, I’ve started getting a feel of what the Frequently Asked Questions are in panel discussions. So I’m just imagining that I’ve been invited to a panel discussion, and the audience is now asking questions, and I’m trying to figure out what answers I’d probably give.

By the way, if any of the readers has a question that isn’t answered here, put a comment stating your question (or send it to me) and I’ll add my answer to that to this post.

How does life as a student in the United States, differ from life as a student in India?

I’d have to confess to very limited experience on both counts: I completed undergraduate studies at Chennai Mathematical Institute (CMI), and am currently doing graduate studies at the The University of Chicago. CMI isn’t the most typical undergraduate institution in India, and I haven’t been long enough at the University of Chicago to udnerstand exactly how the system works.

From what I’ve seen till now, one key difference is the emphasis on assignments and homework. In CMI, we didn’t have a whole lot of regular assignments. Most subjects had at most 3-4 assignments a semester, and there wasn’t a very strict rule about when to submit assignments. There certainly weren’t any graders and teaching assistants: all grading was done by the instructor herself or himself, so the motivation to give assignments was less.

In the University of Chicago, there is a lot of emphasis on regular assignments and regular testing. Apart from the teaching sessions, there are also separate sessions where the teaching assistant helps sharpen the students’ problem-solving skills. In short, there’s more activity. Most of the assignments (as far as I can see) aren’t deep or creativity-inspiring. They’re largely mundane, with a few gems. The goal, from what I can make out, is to give students a lot of practice and a lot of experience in writing solutions.

Moreover, all assignments are graded by the teaching assistant, rather than the instructor. So there is a full-time involvement of two people in teaching a course, rather than of one person. Both teaching and learning is a more hectic and intense activity.

Another difference between life in CMI and life in the University of Chicago is the size and diversity of the place. CMI was largely an “only-math” institute. There were only three subjects of study: mathematics, physics and computer science. Even within these subjects, the number of people doing research and the amount of active research was fairly low. In the University of Chicago, on the other hand, there are researchers of all stripes. There are a whole lot of Ph.D. students. And there are also researchers in other disciplines, and there are a whole lot of facilities apart from those geared at research (including an athletics center, a bookstore, some huge libraries, an office of international affairs, and many others). There are more festivities and occasions and activities outside academics (just as there’s more activity within academics).

Again, I have limited experience in this area. Broadly, I’d say that in undergraduate life, the focus is on learning a lot of different things, on scoring well, on getting a good idea of mathematics, and on developing credentials that’ll impress graduate schools. There’s also a strong element of competition with others in the same year, because there are far more undergraduates than the number of openings at graduate schools. In other words, just as much as it matters how good you are on absolute terms, it also matters (at least, to some extent) how you compare with your immediate peers.

In graduate school, this element of competition is probably less. Of course, the number of post-doctoral openings is significantly less than the number of people who do a Ph.D., so there is still strong competition. However, the competition now is not so much against your immediate peers in your specific institution. Applications beyond graduate studies are fairly specialized, so people with different mathematical interests are likely to be seeking different jobs at different places under different scholarship schemes. Thus, the element of cooperation is probably more.

Graduate school also brings with it a host of teaching responsibilities. Undergraduates are largely responsible or answerable only to themselves. Graduates, who are also involved in teaching duties, have to balance their research with their teaching requirements.

Finally, graduate studies, at the end of the day ,requires a Ph.D. You don’t get a Ph.D. for general knowledge in mathematics. Rather, a Ph.D. is contingent on solving one or two specific problems, usually problems that extend things that have already been studied by the research community or problems that would open the way for new ideas. So graduate studies isn’t just more of the same undergraduate studies. It requires going deep into some area and struggling hard to solve problems, often giving up and returning again.

In what ways did your undergraduate education prepare you for graduate school, and in what ways do you think you were underprepared?

As far as basic mathematical knowledge is concerned, I’d say that my undergraduate education at CMI, combined with whatever I had read on my own, prepared me fairly well for graduate studies at the University of Chicago. Of course, one can look back and say: I wish I had done a course in algebraic topology or if only I had taken a course in functional analysis. But many of my friends who are from leading American univerisities, also hadn’t taken such courses in their undergraduate years, so I do not think I was significantly disadvantaged in a uniform way, as far as coursework was concerned.

That said, I’d say that considering that the CMI course is three years long, while that in most American universities is four years long, there are elements of detail that aren’t covered in CMI courses. Secondly, because we didn’t have to do too many assignments, I didn’t brush up my calculus skills at the undergraduate level. Thirdly, my impression is that CMI’s focus on analysis is low, and this was one of the factors that made me find analysis disproportionately harder.

However, what I value most about my undergraduate life is that it gave me the freedom to explore in mathematics and it was during those years that I started forming my interests. It is possible that if I were subject to a hectic system of assignments, I would not have been able to explore so much and develop the specific interests and viewpoints within mathematics, that I did. On the other hand, it is also possible that a more hectic and challenging course structure could have brought out more from inside me. I don’t know.

Undergraduate program(me)s in the United States are four years long, while the typical duration for a B.Sc. in India is three years. Is this a disadvantage?

It really depends on the specific nature of the undergraduate course. An honors course that covers most of the prerequisites needed for graduate school, is good enough to satisfy most US graduate schools, though some of them have a strict four-year requirement.

However, it is true that B.Sc. courses offered at most institutes in this country do not cover enough material to start a Ph.D. with. In that case, doing a M.Sc. is advisable before applying for a Ph.D.

Why did you apply for graduate studies to the United States? How do options in the United States compare with those in India?

I was specifically interested in the University of Chicago, though I did apply to many other graduate schools in the United States (some of which turned me down as well). My interest in the University of Chicago arose because of two factors: the overall reputation of the University, and the presence of two group theorists: Professors Alperin and Glauberman. It didn’t hurt that some of my seniors and other acquaintances were already at the University and had given me positive reviews of the place.

On the whole, I’d say that if you can do it and are keen on pursuing doctoral studies, then universities in the United States are a good option. Being in the United States isn’t a magic cure. But there are certain factors about the United States universities that set them apart. First, a reasonably large university in the United States will have people doing research in diverse topics, so there is enough scope to interact with a lot of different styles of doing mathematics. Second, there is a culture of hardwork and commitment, and the opportunity to acquire skills in teaching, presentation, research, and forming mathematical communities. Third, there are also good opportunities to interact with other departments.

In India, Tata Institute of Fundamental Research is a place where high-quality research is done in mathematics and a number of other science subjects. It’s small compared to a place like the University of Chicago, but may be among the best choices in India. The Indian Institute of Science has a greater diversity of subjects, but its mathematics department is relatively smaller. From what I’ve heard, the mathematics department is expanding. The Institute of Mathematical Sciences is a good place to pursue doctoral studies in mathematics. The other subjects here include physics and computer science, and all three departments are fairly good.

Nonetheless, an opportunity to study in a top-range or mid-range US university for a Ph.D. can prepare one better for the challenges of research (in my opinion).

What are the pros and cons of pursuing undergraduate studies in the United States as against in India?

The pros:

• More rigorous and challenging coursework
• Greater interaction with other disciplines, the option of doing a double-major, the option of studying a lot of subjects on diverse areas
• Greater opportunities for undergraduate research, and the opportunity to interact with cutting-edge researchers during day-to-day coursework

Cons:

• Making a move to a new country at a younger age could be harder. There is less interaction with the family, and it could be harder to make friends.
• If you ultimately plan to return to India, it’s also important to understand the structure of research and education in India. My stint as a student in CMI gave me a fair idea of what goes on in India. So I’m better equipped to return to India at a later stage.
• The coursework could get really hectic, and making a transition between the exam-based style of learning in India and the assignment-intensive style of learning in the United States could be hard.

None of the cons, however, is totally unavoidable. In other words, by being aware of the possible cons, one can plan to minimize their effect. Thus, if you plan to go abroad, start preparing for it psychologically, physically and mentally a year in advance. It’s not the quantity of preparation as much as the mental transition needed.

Similarly, if you want to go abroad for undergraduate studies, but want to stay in touch with the research opportunities in India, this too is not hard, if you’re aware of it.

The most important thing to remember is that just after high school, we could be pretty impressionable, and may get swept off by the assignment-intensive style of doing things, losing out some of our skills and approach. So it is important to keep in mind that there are a wide variety of different ways of learning and approaching a subject. While joining a University, one must abide by and work with the rules of the system. But that doesn’t necessarily mean that one needs to embrace that system in every way. One can try to keep the best from all the different systems of learning one has seen and learn in a way that is most suited to one’s personality.

Is the money you receive sufficient to meet your expenses in the United States?

The University of Chicago offers a fairly generous stipend, which is considerably more than enough to meet day-to-day expenses. Though stipends vary widely across Universities, mathematics departments by and large offer stipends that can cover living expenses reasonably well, and probably allow you to save money.

For undergraduate studies, the situation is somewhat different. Students from India usually get tuition waivers, but scholarships that more than cover the expenses are relatively rare. I don’t know much about this, though.

How does the United States compare with Europe?

Europe and the United States differ in various respects, but again I don’t have much firsthand experience of studying in a European institution. One difference is language. A second difference is probably culture. The core American system is based on hardwork and the testing pattern is usually assignment-based. Students are actively involved with teaching and learning. In Europe, the system of scholarships, stipends and other stuff works differently.

If there was one thing you wish you’d known before beginning graduate studies in the University of Chicago, what would it be?

It would probably be that even though the system of evaluation and the structure and setup is different here compared to what I’m used to, the fundamental values remain the same. These are the fundamental values of being sincere, hardworking, creative, and cooperating with others. The fundamental principle is to be honest to oneself, to have faith in one’s abilities, and to have fun and find one’s equilibrium in a new climate.

I cannot really say I suffered here; I did pretty well in the first two quarters. But during the first quarter, I was somewhat stressed because of the expectations of the new system with three assignments a week. It took me some time to come to terms with this and to find joy and fun in my day-to-day activities. I’d encourage everybody who goes to a new learning environment to not be blown over by the superficial differences and to know that in the end, it is good fundamental values that triumph.

## March 28, 2008

### Glauberman conference

Filed under: Places and events — vipulnaik @ 11:44 pm
Tags: , ,

Today was the last day of a five-day long conference on group theory. This was the Glauberman conference, held right here at the Mathematics Department of the University of Chicago. The conference was in honor of Professor George Glauberman, a leading group theorist who’s proved results like the ZJ-theorem and the Z*-theorem. Prior to this conference, I’d heard of mathematical conferences and read books with proceedings of these conferences, but I didn’t have any experience of attending a conference. So I was very eager to attend this one. “Conference” can have many meanings. The Glauberman conference was primarily a series of lectures by different mathematicians on different topics. In fact, most of the lectures were on very specific topics, they were short (about 30-35 minutes) and there wasn’t a unifying theme to the talks. I didn’t follow too much of the content of the talks, primarily because of the fast pace and the large number of talks. But it was a great experience to meet people from across the world (group theorists from Europe and Japan had also come). I’d read books writen by some of these people, and had also corresponded with some of them, so it was nice to see them in person (though the schedule was too hectic to interact more with them). I got an idea of the notational conventions that were followed in group theory. I learned that the default convention in group theory is to make elements act on the right (especially when writing down cumbersome commutators and expressions to simplify) rather than on the left. More importantly, I got to understand some of the important research themes in the subject. One research theme is around a collection of conjectures intended to understand better the relation between the representation theory of a huge group, and the representation theory of “local” subgroups (small subgroups that occur as normalizers of subgroups of prime power order). The first conjecture in this regard was by McKay. McKay conjectured that the number of irreducible representations of $p'$ order of a group equals the number of irreducible representations of $p'$ order of the normalizer of any $p$-Sylow subgroup. Many modifications of this conjecture have been proposed by Alperin, Isaacs and others. In a similar but somewhat different vein, there’s the Glauberman correspondence, that gives an explicit bijection between the representations of a huge group and a smaller subgroup. This, too, has spawned a number of related thoughts. There were some talks in the Glauberman conference that focused on some of the applications and results inspired. Professor Bhama Srinivasan, who gave a lecture about some correspondences involving linear groups, told us that the whole spectrum of conjectures had been summarized as “I AM DRUNK” where the letters stood for the initials of the people who had come up with variations of the McKay conjecture. (I, A, M stand for Isaacs, Alperin and McKay; I forget all the other letters right now). Another important theme was the theory of “replacement”: replacing a subgroup satisfying certain, weaker properties, with a subgroup satisfying certain, stronger properties. Thompson was the first person to come up with replacement theorems, and Professor Glauberman has published a number of recent results in that regard, making good use of the ideas behind the Lazard correspondence. One of the interesting results was mentioned by Professor Khukhro, who was inspired by Professor Glauberman’s replacement theorem to prove a very general result that works for all groups: a normal subgroup of finite index can be replaced by a characteristic subgroup of finite index, and satisfying the same multilinear commutator identity (so for instance a normal nilpotent subgroup of finite index can be replaced by a characteristic nilpotent subgroup of finite index). Group theory’s recently winning the attention of people in topology and category theory. During the classification of finite simple groups, there were some “candidates” for finite simple groups that never materialized into actual groups. however, there was a lot of data in these cases to suggest that a group exists. later, it was discovered that one could define an abstract notion, called a fusion system, and that every group gives rise to a fusion system, but there are fusion systems that don’t come from groups. Fusion systems are something like a piece of consistent data that could have come from a group, but on the other hand, may not. Some recent work has gone on into find out what are the fusion systems that do not come from groups, and how one can judge whether a given fusion system arises from a group. The talks at the Glauberman conference weren’t directly on these basic concerns, but on some related research. This included talks by Cermak, Bob Oliver, and Radha Kessar. There were also talks related to classifying and making sense of the $p$-groups (groups whose order is a power of a prime). Classifying $p$ groups is a tricky proposition: it makes sense only if we decide what it means to “classify. Professor Leedham-Green gave a talk on classifying $p$-groups by coclass. All the talks were 30-35 minutes long. Some of them used the chalkboard, others used laptop-based presentations, and yet others used transparencies. In fact, the ones using transparencies used two projectors, with one projector used to show the “previous” transparency for reference. It was good fun.

## January 3, 2008

### Eat pizza and a math speaker

Filed under: Chicago — vipulnaik @ 8:52 pm
Tags: ,

In an earlier post, I had described the system of student talks that I had initiated at Chennai Mathematical Institute, I’ve now passed out of CMI and the student talks are still continuing; in fact, they are flourishing better than they were in my time, thanks to the efforts of Swarnava and Kshitij.

Student talks at CMI were a small-town affair in my time: audience sizes ranged from 3 to 10, the speaker (usually I) would wait for all the people he/she knew would attend, before beginning, and the talk had no scheduled end time there was an estimated talk duration but nobody was accountable for it). Talks were schedule on arbitrary days, at the convenience of those who were interested in attending. The talks were usually delivered in a seminar hall, which had a seating capacity of around a 100 people, and I often used slideshows. With an audience of only 4-5 people in the huge hall, it was almost like a luxury event.

The student talks (dubbed Pizza Seminars) at the University of Chicago are a completely different affair. The first major thing is that the lure of mathematical knowledge is not the only incentive for attending: there’s free pizza too, and the talk is scheduled during the lunch hour, which, for the first-years, is the break-time between two lectures. Many of us, tired from the previous night’s assignment slog, used to grab pizza, finish it, and fall off to sleep.

(The pizzas are not funded by the speaker; they’re funded by the Math Department and the Physical Sciences Division of the University).

Secondly, the talks are held on a fixed schedule: once a week, at 12:30 p.m. on Wednesdays. This isn’t surprising; if pizzas are being offered, one can’t schedule an unlimited number of talks based on whims and fancies. Speakers have to start on time, break on time (for a second serving of pizza) and end on time.

Thirdly (and I wouldn’t say this is independent of the first reason) the talks are much better attended. The Barn, which can seat around around 50-60 people, is nearly full for every talk.

This leads to a lot of differences in the way the talks are conducted. In our undergraduate institution, people were often quite passionate on the topics they were talking about; here, the talks are largely viewed as a supplement to the pizza, and speakers, even the good ones, appear more indifferent to the impact they make on the audience. With a long line of speakers and only one talk per week, it’s very different from the situation in CMI where a talk slot was actually fixed based on the convenience of those who regularly attended. Finally, with such a huge audience, it’s much easier to get lost in what the speaker is saying, although there are a number of spirited interruptions to the Pizza Seminar in the first five minutes when the audience hasn’t given up hope.

An example can be the talks I myself gave on the same topic, one in CMI, the other in Chicago. The topic was extensible automorphisms. In the talk I gave at CMI, I spent a large amoun of time defining groups, inner automorphisms, extensible automorphisms, and developing machinery of representation theory as well as some leading ideas. The audience was much younger, and I’m not sure how much they understood of the talk, but there was a slow-world air about it. My talk in Chicago was a light-hearted but quick-styled affair; I jumped from here to there, throwing in some wry humour at various points, and trying to give a quick peek into the topic to a significantly smarter and more knowledgeable, but on the whole more preoccupied and less interested audience.

A more fundamental difference between the Pizza Seminars in Chicago and the student talks in CMI, however, is the backdrop. In our undergraduate institution, students often have little or no opportunities to teach or explain in a formal setting; graduate students in the University of Chicago, on the other hand, get to teach or assist in teaching regular courses. Thus, there is hardly that much novelty value in addressing a large audience. Further, there is not much one can do to “teach” graduate people in one hour. Student talks in CMI were actually teaching and learning opportunities, even if what was learned was eventually forgotten.

A few weeks ago, I asked the current coordinators of student talks at CMI, about the possible directions these talks might take if pizzas were offered at each talk. He said that attendance would certainly rise, because people cared more about pizzas than about math. Whether that is a good thing or a bad thing is debatable. The debate here is not about the health impact of pizzas, but rather about the issue of what the goal of talks is, who the target audience is, and what kind of value (over and above the pizza) is to be imparted to the audience.

## August 30, 2007

### Setting off to Chicago

It’s a long time since I last posted on this blog. The last two months, since I returned from Paris, have largely been holiday time for me, and I’ve been doing some miscellaneous stuff to prepare myself for the next important phase of my life. On September 8, just ten days from now, I will board a plane to Chicago, to begin my five-year doctoral programme in mathematics at the University of Chicago.

The first year of the programme at the Universty of Chicago is mainly compulsory coursework. There are three quarters (each three months long) and three course sequences (Algebra, Analysis and Topology). In each quarter, there is one course from each sequence. So a total of nine courses for the first year.

Chicago differs in this respect from other graduate schools. In some graduate schools like Princeton, there is no well-defined framework of compulsory courses, rather students have to pick and choose their courses from a set of recommended courses and prepare themselves for examinations at the end of the first year. From what I can infer, the emphasis in places like Princeton is to get people started on research-like work from a fairly early stage. The pressure to publish thus begins in the first two years itself. In Chicago, on the other hand, there is no pressure to publish; the first few years are meant to strengthen the fundamentals in various areas of mathematics and research is intended for later years.

A couple of months ago, I received an email from Peter May, addressed to all the incoming graduate students, about coursework for the first year. I found that a lot of the material in the courses, particularly the Analysis sequence, was completely unheard of, and thought I should probably start reading up for it. However, I started reading up measure theory and analysis only recently, and am finding it somewhat hard for now. This is probably the consequence of not having done any courses in measure theory and not having a good analysis background. I hope that by contrast, my somewhat better background in algebra will prove an asset to me for the algebra courses, particularly the courses in representation theory and groups. Areas where I have a little, but not a very good, background, are algebraic topology, commutative algebra, and algebraic geometry. In these areas, I hope to keep reasonable pace with the coursework, though I probably will not find it too easy.

I think that the course-based structure for the first year at Chicago will definitely be a help to me so that I can get up to scratch in all important aspects of mathematics. More importantly, I will be able to overcome the fear and reluctance that I currently have with certain kinds of proof techniques and terminology (particularly that of analysis). Another advantage of such a structure is that I will automatically get an opportunity to interact with a number of Chicago faculty members in all branches of mathematics, something I may not be able to achieve of my own initiative. Further, I will also get to interact with my fellow graduate students in and outside the classroom.

In my second year at Chicago, I will be expected to write a paper in a topic of my choice, acquire a working knowledge of a language other than English (I’ll probably choose French, given that have already picked up some French) and submit a master’s thesis. During the second year, I will also be functioning as a Teaching Assistant for an undergraduate math course.

From the third year onwards, I will be expected to start doctoral work full-force, and simultaneously I will need to teach a course of Freshman Calculus. In Chicago, as in many American universities, all freshmen (incoming undergraduate students) need to study one calculus course, irrespective of their stream of specialization. The job of teaching these courses is assigned to doctoral students in the mathematics department.

Currently, it is too early for me to think of questions like what topic I will choose for my thesis, who my thesis advisor will be, how many years it will take me to complete my thesis work, and whether I want to continue to a post-doctoral position in mathematics after that. I do have some ideas and preferences on these counts, but it is only after I go to Chicago and observe the work environment there, and assess my own research abilities in that environment, that I can take the correct decisions. For now, my focus is to equip myself to get the best out of my first year, and to understand the temperament and qualities needed for research, through close observation.

## June 21, 2007

### The ENS — wrapping up

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

### 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.

### 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

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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

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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