What Is Research?

May 19, 2008

Spring quarter

Filed under: Regular updates — vipulnaik @ 11:20 pm

This is a bit late in the spring quarter to be blogging. Already, seven out of the ten weeks of “education” for the quarter are over, so what’s left is three more weeks of learning, then one final exams week (and we don’t have any exams, so that’s basically free time, except possible one take-home). So how’s this quarter been different from previous quarters.

In some important respects, this quarter has been considerably more relaxed than previous quarters. The main gain this time was that the complex analysis instructor, Professor Wilhelm Schlag, decided not to assign us the regular weekly homework assignments. Instead, he’s providing us class notes with a list of problems with solutions. The idea is that we are supposed to attend class, read the notes at home, and solve the problems he’s given. There was a take-home midterm and there will be a take-home final.

The other two instructors are giving weekly homeworks — Professor Webster gives regular weekly homeworks in differential geometry, but these tend to not be too long. Professor May gives weekly assignments, too, some of them with hard questions, but the first-year office has now achieved maximal skill at doing assignments using the maximum of teamwork and with the minimum of individual strain.

Assignments in the two subjects are due on Wednesday and Friday, which leaves the weekend relatively unfettered.

So it’d seem like I’ve been goofing off, right?

Well, not quite. There’s always a lot of stuff to do in the coursework, even without assignments, though I don’t always do it. Still, it’s there to fill my plate when it gets too empty. But I’m also working on some other things; for instance, I’ve been working on setting up a permanent home for the subject wikis I blogged about earlier (I’ll get back to that in a later blog post). That’s been the main occupation, but I’ve also been doing some general reading and learning that’ll help me in later life. Most pertinently, I’ve decided on my advisor. I plan to work with Professor George Glauberman. He’s given me some material to read (from Daniel Gorenstein’s book) and I’ll be working on that over the coming week.

So how’s spring quarter been different from the preceding quarters?

I remember that during the fall quarter (September – December), I was pretty keyed in to assignments; they were the mainstay, and most of the free corners of time that I’d get, would be devoted to trying to solve assignment problems. This changed a bit towards the end, when some aspects of the subject matter themselves became more interesting, and I prepared some write-ups in topology and algebra (links can be found from my Chicago home page). During the winter quarter, I was less hyped about assignments, though the work continued to be hard. By this time, the first-year office had evolved a more stable strategy of assignment-solving, too, which meant that people solved problems together in optimal-sized groups (the optimal size being four), wrote down the solutions on the chalkboards in the first-year office, enabling easy copying.

During the winter quarter, I also started working a little more on academic stuff that wasn’t directly related to the immediate needs of coursework, which includes the subject wikis in topics like group theory, topology, and commutative algebra. I also started reading on some other things that weren’t directly related to mathematics and academics, and spent some time (though not a very significant amount of time) watching TED videos. I also started going more regularly for daily jogs, something that had suffered during the Fall quarter.

In the spring quarter, I’ve been more relaxed, and have continued the tradition of working hard but not in an assignment-dictated or course-determined fashion. I spent a reasonable amount of time with various parts of the coursework, but I’ve also been thinking of other things. Partly, this could be attributed to fatigue with the assignment system. Let’s hope it doesn’t backfire and I get smoothly through the remaining three weeks of assignments and coursework.

March 8, 2008

Tryst with functional analysis

It’s the end of the ninth of the eleven-week winter quarter, and the next two weeks are probably going to be fairly hectic: we have examinations/final homeworks to submit in all subjects, and I’m guessing that from tomorrow onwards, work on these will begin full-force. So I’m taking a little time off right now to describe my tryst with functional analysis so far.

During the first 1-2 weeks of the functional analysis course taught by Professor Ryzhik, I was enjoying the material, more or less keeping pace with the material, and also reading ahead some topics that I thought he might cover. However, from around the third week onwards, the nature of topics being covered in the course changed somewhat and I started getting out of sync with the material. Then came an assignment with problems that I had no idea of how to solve. Eventually, solutions to these problems were found in an expository paper by David (one of my batchmates) and the first years worked out the details of the solution on the chalkboard.

At the time, I was feeling tired, so I didn’t try to keep pace with and understand all the details of these solutions. I wrote down a reasonable bit of them to muster a decent score on the assignment but I didn’t internalize the problem statements (I did have some ideas about the problems but not from the angle that Prof. Ryzhik was targeting).

So, in the next week’s problem set, I wasn’t able to solve any of the problems. This wasn’t because the problems were individually hard (though some of them were) but because even the easy problems needed a kind of tuning in that I hadn’t done/ I learned of the solutions from others and understood enough of them to submit my assignment, but they hadn’t sunk in. At the same time, I was handling a number of other things and I didn’t have a clear idea of how to proceed with studying analysis.

Some time during this uncomfortable period with the subject, I remembered that the previous quarter, I had overcome my discomfort with noncommutative algebra by writing Flavour of noncommutative algebra part 1 and Flavour of noncommutative algebra part 2. Noncommutative algebra differed from functional analysis: in the former, I was reasonably good at solving individual problems but just hadn’t had the time to look back and get the bigger picture. In functional analysis, I didn’t start off with a good problem-solving ability or an understanding of the bigger picture.

Nonetheless, I knew that trying to prepare a write-up on the subject was probably the best way of utilizing my energies and probably a way that would also be useful to other students, which could partly be a way of contributing back, considering that I hadn’t solved any of the recent assignment problems. Moreover, it was something I knew I’d enjoy doing and I hoped to learn a lot from. So I got started. The first attempt at preparing notes was just aroudn the corner from the mid-term. I got a lot of help from Rita Jimenez Rolland (one of my batchmates) who explained various parts of the course to me as I typed them in. (Here’s the write-up).

However, after the examination (where I didn’t do too well — notes are more useful if not prepared at the last minute) and as I learned more and more of the subject, I felt that it’s good to restart the notes-making process. I brainstormed myself about what kind of write-up would be most useful. Instead of just trying to cover whatever has been done in the course, I tried to look at the problems from a more basic angle, like: what are the fundamental objects here? What are the things we’re fundamentally interested in? I also brainstormed Mike Miller, who provided some more useful suggestions, and I got started with the write-up.

Preparing the analysis write-up hasn’t been plain sailing. The problem isn’t so much lack of time, as it is lack of richness of engagement. When I’m working on my group theory wiki or writing this blog entry, or doing something where I have a very rich and vivid idea of what’s going on, every part of my mind is engaged. There isn’t scope for distraction or going lax, because I’m engaging myself completely. However, when writing functional analysis notes, I faced the problem of my own ignorance and lack of depth and ideas in the subject. So, when I got stuck at something, I didn’t have enough alternate routes to keep myself engaged with the subject. The result? I kept distracting myself by checking email, catching up with other stuff, and what-not.

The contrast was most striking some time about a week ago. Through one hour of interrupted and not-very-focussed work on the functional analysis notes, I was getting somewhat frustrated. On a whim, I decided to switch to working on the group theory wiki. I did that, and was surprised to observe that for the next one hour, I didn’t check my email even once.

The complete concentration on the subject isn’t merely explained by the fact that I like group theory more, or am better at it. It is more the fact that I can see a bigger picture. Even if I’m concentrating on a couple of trees in the forest of group theory, I can see a much larger part of the forest. But when working on a couple of trees in functional analysis, all I can see is those and a bunch of other trees. So distractions find their way more easily.

I consider this illustrative because we often think of concentration as a kind of tool of willpower. True, the exertion of willpower is necessary to concentrate at some times (e.g. to pull myself back from the group theory wiki and back to functional analysis). But more fundamentally, I think it’s the intrinsic ability to see something as very rich and beautiful and to keep oneself completely engaged, that matters. Do determination and hardwork play a role? Yes, they do, but they do so because they help build that internal richness. Which explains why I love writing so much: in a number of areas, writing allows me the most to explore the inner richness. And I think this is a factor in explaining why, although many different people work hard, there are not so many who, at the end of their hardwork, find the work enjoyable. That’s because most of us use a very small part of the tremendous hardwork that we put in, into creating an internal richness that can engage us better.

What about functional analysis and me? Do I see the richness in functional analysis yet? Not to the level that’d help me cope very effectively with the course, but yes, I do feel a lot better about the subject. And I think the new notes on function spaces, even though they may seem amateurish right now, do indicate some of the insight and richness that I have gathered over the past few weeks. Let’s hope I can augment these notes in the coming days to a level that really gets me prepared for the examination!

February 4, 2008

Quarterly progress

When I started life this quarter, I had determined that it would be more enjoyable than last quarter, with less paranoia about assignments, more fun in the learning process and a cooler and calmer perspective to life. Things have been going fairly well in all respects.

Probably the first difference is that I’m much calmer about assignments, even when they don’t get done or are left right for the last minute. Providence also seems to have helped me; the assignments are (by and large) shorter, though there are some exceptions and I have to sometimes do a hasty last-minute job. But then, I had to do hasty jobs last quarter too; the difference was that assignments occupied much more mental space so that I couldn’t concentrate on doing the things that I liked.

One thing I’ve been experimenting with is wikiing while I work, and that means that as I’m learning stuff, I’m constantly thinking of how it can be organized on and integrated with the wikis that I’m working on. I’ve been augmenting the Commutative algebra wiki as I go along. This hasn’t been instant magic, because I don’t have the kind of feel for commutative algebra to immediately see how certain facts can be organized, but it means that I’m thinking of the subject in a way that’s not just limited to assignments. The wiki’s also becoming a useful, no-nonsense, reference point for me, and a convenient way to augment my memory and intelligence.

Differential topology is very interesting, and while studying it, I have to keep updating two wikis, the Topology Wiki, and the Differential geometry wiki, which often cover similar stuff from slightly different perspectives. I had worked quite a bit on organizing the topology wiki over the winter so every new thing I want to say seems to have a nice place to put it, and it seems to be not too far when the wiki will start exhibiting the kind of beautiful self-organization that I’m seeing in the group theory wiki.

The fact that there’s a course on local analysis in finite groups keeps me very happy. Although I can’t devote too much time to group theory while in the midst of all my compulsory courses, this course at least keeps me on track in the subject. It’s fascinating to see in formal proof all the things that I have picked up from textbooks and miscellaneous papers. I hope that I can really work out on the Group Properties Wiki soon, though I keep augmenting it from time to time. It’s looking more and more beautiful.

I’ve also been discussing some ideas in group theory with Professor George Glauberman, the instructor for the course on finite groups. Again, I plan to pursue them more later on.

The functional analysis course isn’t going as well as I’d hope, but I’m still having fun trying to follow the ideas. The problem for me is that the topics and direction are changing rapidly. Assignment-solving had a huge collective component last week (in other words, for many problems, I couldn’t figure out the solutions even after I wrote them). But at least there are some things in the subject that I’m learning. I am trying to wiki things out there on the measure theory wiki, but since I hadn’t set it up and structured it, what I add ar ejust isolated articles, and there’s no bigger picture emerging.

There’s also a course by Professor Victor Ginzburg on semisimple groups and geometry that I’m attending. For the first time, I’m seeing proofs (although more on the line of outlines of proofs) for statements on algebraic groups. This isn’t my primary focus area but it is something I’d like to understand well and Professor Ginzburg’s approach is interesting and his excitement is infectious. Unfortunately, I’m not getting to spend time on this outside class. It does remind me of some things I’ve played around with, like APS theory and the log category, and I hope that with a better understanding of semisimple groups I can come back to these and put more life into them.

How much I’ve learned this quarter remains to be seen, but I’m definitely enjoying it a lot.

January 3, 2008

One quarter down

Filed under: Regular updates — vipulnaik @ 8:13 pm

This is my first post since coming to the University of Chicago for the mathematics Ph.D. programme. I offer no excuses for the long delay — the main reason was that the past quarter at the University if Chicago was nothing short of hectic, and there were a lot of things I wanted to catch up with in the one-month winter break that followed (there’ll be more on that in subsequent posts). Let me try to give an idea here of the graduate work at the University of Chicago, and how it differs from my undergraduate experiences.

In the first year of the Doctoral Program at the University, we have absolutely no teaching duties; all mathematics students are on a University fellowship, and the main task of the first year is to get through the compulsory courses. There are three course sequences: Algebra, Analysis and Topology/Geometry, with one course of each sequence in each quarter. A “quarter” is an eleven-week term, and there are three quarters in the year (Fall, Winter and Spring). There’s also a summer quarter, which is the time for doing summer study, or freaking out.

Many universities in the United States run on the quarter system; examples other than Chicago are Northwestern University and California Institute of Technology. Others run on the semester system, which is closer to the system in India: two terms, each approximately 16 weeks. The quarter system means shorter, and more intense courses, and more frenzy, but it also means that you have to tolerate a course you don’t like for that much less time.

My first quarter at the University of Chicago was from September 24th to December 7th; the final week was examination week, so we effectively had ten weeks of instruction (with a short break for Thanksgiving). In this quarter, there were three compulsory courses. Each course had three lectures per week (three seems to be the favourite number here), one each on Monday, Wednesday and Friday. An assignment (whose solution was usually around 6-15 LaTeXed pages) was due in each subject each week, and the submission dates for the three subjects were Monday, Wednesday and Friday. Apart from coping with the material covered in lectures, the primary focus was thus getting through assignments.

The general expectation, from what I understood, is that everybody is expected to submit assignments on the due date, with complete solutions, and students are strongly encouraged to discuss solutions with others if they are not able to solve the problems themselves. The place where these discussions took place was a musty underground dungeon euphemistically called the first-year office, where all the first-year students had appropriated desks. Often, the day before assignments, the boards would be full of solutions or key ideas for solutions, with people hopping around and explaining the solutions to each other. It was not uncommon for students to actually take down notes on their laptops as solutions were being explained. The first-year “office” would usually be up and running till late at night, and usually till early morning, before the submission of the assignment.

Assignments filled up so much of our mindshare that keeping track of what was taught in the classes was often a secondary, or even irrelevant, concern. Nonetheless, well-designed assignments forced us to go back to material covered in class and thus led to a high probability that we assimilated well the topics that were covered in class, as well as those that were glossed over or given short shrift.

The Algebra course was taught by Professor Victor Ginzburg, a well-known person who works in (as far as I understand) a broad gamut of noncommutative geometry and representation theory. Algebra was my preferred subject when I came to Chicago, and the first few weeks in Professor Ginzburg’s course were quite pleasant, although his teaching style was different from what I was accustomed to. Towards the later part of the course, Professor Ginzburg switched to noncommutative algebra, a topic which was largely new to me, and I felt increasingly frustrated at my inability to take out time to study the topic and having to cope with the assignments one at a time rather than getting a bigger picture of what’s happening. It was finally around the time of Thanksgiving that I decided to channel my frustrations into something positive, and started preparing notes titled “A Flavour of Noncommutative algebra”. These notes were such fun to write that I actually started enjoying the beauty of noncommutative algebra, and many of the pieces which Professor Ginzburg had mentioned in class or given us in the assignments, started fitting together. I enjoyed them so much that I even wrote up a Part 2 following the first part, and passed on the notes link to my batchmates, some of whom gave a number of useful comments that helped me augment the notes. For those who’re interested, here are Part 1 and Part 2 respectively.

On balance, algebra could have been a more enjoyable experience for me than it was. One of the reasons why I found it hard to enjoy or appreciate was that I had a lot of previous notions and ideas in algebra, and so whenever something was covered, I would always feel that it would have been better viewed in this way, or that some essential point was missed. This led me to resent the subject in a way that was unnecessary, and I would probably have done better to get started with preparing my notes and trying to see the new perspectives and ideas earlier on, rather than wait for Thanksgiving time to catch up.

Analysis was taught by Professor Gregory Lawler. Being a probabilist by profession, Professor Lawler mingled in a lot of probability with the measure theory and analysis, which made it more interesting as well as harder. Although the level of material we covered in the course was not conceptually too hard, the assignments were demanding, specially for me, since I had no prior experience of solving analysis problems. Analysis was the subject where I gleaned the most from the first-year office, and had the least amount of insight from within, and it is probably the subject where I will need to put in the maximum effort to get the rhythm in future quarters.

One of the mistakes I made in analysis (and which I hopefully will not make in future quarters) was that because the first few weeks were light, I did the assignments and didn’t think about the subject further; I didn’t take the opportunity to familiarize myself with subject material in later chapters. I should have realized that given my poor grounding in analysis (compared to many batchmates who had done a graduate course in real analysis earlier), reading ahead would be profitable. But then, one lives and learns (hopefully!).

Algebraic topology, taught by Professor Madhav Nori, was for me the most fascinating new thing. In the first couple of weeks, I was not following too much, and moreover, I was making a lot of careless errors with the subject; long exact sequences of pairs, Mayer-Vietoris, and the like seemed mumbo-jumbo and I’d often mix the pair with the subspace. Again, it was somewhere around the fourth or fifth week that I decided to take some time off and study the subject properly. It was just a single day off; which cost me significantly in terms of assignments, but which gave me added confidence in the subject and helped me to then build a love for the subject, and a reasonable degree of intuition. It was also an occasion for me to revisit point-set topology, and to appreciate many of the subtle points in point-set topology that I had started exploring long ago, and then abandoned.

This first quarter at Chicago was extremely different from the academic atmosphere I was used to in my undergraduate institution. In most of the courses there, we had little homework; there were plenty of courses with no assignments, and even those courses which had assignments didn’t have more than three. There was only one course where we had weekly assignments, and this course was taught by Dr. Amritanshu Prasad, who did his Ph.D. at the University of Chicago. But even his assignments were much shorter than the average assignment we got here, and we had to do three in a week here!

What I learned from this first quarter most was the importance of taking time off in different senses. Firstly, there’s taking time off from studies altogether, which is the obvious meaning, but secondly, there’s taking time off from assignment work and trying to get back and view the broader picture. In some sense, my undergraduate education almost entirely consisted of taking time off, so I never had to put in an effort for it; but here, taking time off requires a deliberate effort.

It is hard to debate (and I have too little experience to even offer consolidated opinions) on whether a system with a lot of homework, as in the University of Chicago, prevents people from taking time off and getting the bigger picture. However, I must say that most people, even in the absence of a heavy workload, find it hard to stand back and get the bigger picture. In my undergraduate institution, the lack of homework pressure, while ideal for people who wanted to take time for their own study and exploration, also meant that people could simply goof off and treat the semester as vacation time: something which is rampant in our undergraduate institution, at least for people who are smart enough to get by the exams with a reasonable amount of effort. Secondly, one can take time off and get the bigger picture only after toiling on all, or at least some of, the little details, and well-constructed assignments give this opportunity.

It falls on me to integrate the various approaches and try to evolve a method for handling my studies which is not only most efficient, but also most stress-free and enjoyable.

August 30, 2007

To be set aside

Filed under: Regular updates — vipulnaik @ 7:37 am

Three years and one month ago, I was at the start of a phase of my academic life: undergraduate study. When I joined Chennai Mathematical Institute for undergraduate study, I was very sure that my future lay in doing study and research in mathematics. My confidence had been boosted by fairly successful performance at the International Mathematical Olympiad (two silver medals). I had also eagerly started reading higher mathematics books, including two books in abstract algebra, and two books in topology. I was eager to learn as much as I could in the coming three years at CMI, and to pave the way for further studies in mathematics.

Now that I have completed my three years at CMI, and am about to start off with a doctoral programme in mathematics at Chicago, I can probably declare myself successful in what I set out to do. In these three years, I have learnt a lot of mathematics, although there are many branches of the subject where my knowledge is below par, and I intend to focus on these over the coming year. Yet, in some ways, my path of learning mathematics was not the way I had envisaged. My term-time and holiday-time learning in mathematics was very different from the way I had studied subjects in school, different from Olympiad mathematics as well. For school study, all I typically did was to sit quietly, read a book, make a few notes, solve a few problems, and I would have “studied” a particular topic.

But after I joined CMI, I found that I had little patience, and little need, for that kind of approach. In fact, a whole lot of college life and hostel life was so full of distractions and other things around that I could not really sit quietly, make a few notes, solve a few problems and move on. Rather, my study of mathematics was disorganized and fragmented — read a bit here, a bit there, juggle it in the mind some other time, and keep failing to understand, but moving on.

My ways of learning the subject were unbalanced and I was not following any particular book, so there were lots of things I would miss out, simple things I would forget, and complex things I would grasp. I also found myself undervaluing, and ignoring, the importance of being “bright” and “sharp” and being a good problem-solver. Instead I was more and more fascinated by the idea of reading a bit here, a bit there, building a grand and beautiful picture, most of it on very elementary proofs but big in complexity because of the large number of building blocks needed.

This eccentric approach towards learning, this obsession with doing things my own way, led me after some time to severely doubt whether I was really enjoying and doing well with mathematics. I found that, looking back on my past few months of work, I could not list too many new things I had learnt. I also started doubting whether mathematics was really as enjoyable and worthwhile as I had initially thought it to be. Further, I started doubting both the utility of mathematics and my own ability to pursue it with a sense of discipline. I feared that I would not have the discipline to study mathematics in an ordered, systematic way.

These doubts gripped me very strongly in the beginnig of my second year. There was also the problem of some courses that I did not enjoy — I skipped classes in those courses, could not bring myself to study those subjects.

Towards the later half of my second year, however, I decided that the study of mathematics is the best hope I have, and that it is the only thing so far that I have learnt both to be good at and to enjoy. Thus, I started re-engineering my life in my fourth semester, and this included determinedly attending all courses (even if I felt too tired) and trying to read up and study new aspects of mathematics.

At the beginning of my third year, after a lot of dithering, I decided to apply abroad for after completion of undergraduate studies. Surprisingly, I found that my sense of discipline was very much there — I was able to study for the general GRE verbal and essay part, the TOEFL and the subject GRE in the same ordered and systematic manner as I used to study in my school days. I also found that I was able to juggle that well with my coursework. In the final semester, I expored the creation of a group theory wiki, and did many other experiments, while reading up and learning new mathematics and taking up a number of extra courses.

Now, as I am on the brink of a new phase of my life, I feel confident that I have the ability and the strength to study and do research in mathematics. However, I am still far from sure that mathematics is the correct choice of long-term career. In some sense, I realize that my youthful confidence that mathematics is my destiny was rooted in ignorance and idealism. Rather, I have replaced it with a more wary attitude where I try, at each stage, to equip myself best for the present and the immediate future while gathering knowledge and resources that will help me in the farther future.

For now, it is time to set aside the past, and get ready for what awaits me.

Setting off to Chicago

Filed under: Chicago,Places and events,Regular updates — vipulnaik @ 7:12 am

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

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.

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.

A new phase of my life in the offing

Filed under: Regular updates — vipulnaik @ 11:01 am

Yesterday night, I was feeling extremely fatigued and worn, and when I stopped to think of what was causing this, I realized that I’ve just been going on and on for days on end without stopping and thinking of where I’m heading.

This semester has been rather light for me (in terms of course load: only five courses) but I have offset that my taking on a number of ambitious and tricky tasks, most notably the Group theory wiki. In addition, I’ve been involved with a teeny-weeny bit of work on things like CMI Spark, CMI Online Programming Contest, Olympiad teaching and training.

I’ve also been giving quite a few Student talks and have been trying to prepare lecture notes for some of my courses such as Lie-theoretic methods in analysis (I haven’t worked on that for a long time, but that’s a different story).

In addition, I’ve been trying to get started on some other wikis, particularly a Differential geometry wiki. And I also plan to get started on my Commutative algebra wiki. I’ve also been trying to work on developing a theory (which again hasn’t been working out, called APS theory, on a wiki devoted to that.

So in the midst of all this (attempted, and so far largely unsuccessful) activity, I have so far forgotten to face up to and think about the new life that lies ahead of me.

To set some background.

I am currently in the third and final year of my undergraduate programme at Chennai Mathematical Institute. Once I complete my B.Sc., I plan to join into a Graduate School (viz a place for Ph.D. in mathematics). Once in Graduate School, I hope and expect to work on research in some important problems in mathematics, probably make some original contributions that’ll earn me a Ph.D., and then, continue with mathematical research.

I’ll be starting on Graduate School probably some time in August-September, and I need to begin the emotional journey towards that. I need to figure out how it’s going to be different from the undergraduate life, what the new pressures and responsibilities will be, and how I can do well in Graduate School.

I also need to look at my more long-term plans and see how Graduate School will fit into them.

Unfortunately, I’ve been too caught up with the things I am currently doing to be able to stand back and stare.

But now, as I write this blog post, I am forcing myself to think hard.

One of the basic differences between my current undergraduate life and life in Graduate School, will be that in Graduate School, I’ll actually be expected to produce results. As in, there’ll be papers I have to read, results I have to create afresh or extend.

Right now, whatever little of papers I read or research work that I try acquainting myself with, I do of my own accord. And I always have the freedom to stop reading when I get bored or when the stuff gets too complicated or when I realize that this is not the right order in which to read things. So, because I don’t really need to get anything done, my efforts to acquaint myself with new research are sporadic and ad hoc.

On the other hand, getting a doctorate, as far as I understand, is not something that can merely be accomplished by knowing a bit about everything, reading a bit here and a bit there. One needs to take a specific problem and make definite progress on it that is worthy of publication somewhere. Trying to make definite progress that hasn’t been made by any person so far seems a daunting task. I mean, what guarantee is it that, however smart I am, I’ll actually be able to find something new and significant in the five or so years I am allotted for my Ph.D.?

From what I have observed of the research world and the way it operates, a crucial component of research is reading and thoroughly studying research papers. Unfortunately, I don’t think I have yet gotten the hang of it — also somehow I don’t seem to enjoy it (of course, it may be too early for me, considering that I am still an undergraduate, but the sinking feeling I get on seeing a long series of pages with abstruse symbols and claims and proofs does depress me). This is notwithstanding a post I myself put on this blog on
How to read research papers.

This is one of the things I need to look into to overcome. So far, I have looked at research papers, but it’s usually for something specific — a definition, a little claim, a particular theorem, etc. The first big attempt I made to study a research paper was under Professor Dipendra Prasad at the Visiting Students Research Programme at TIFR, Mumbai. This was the paper on Lie Group Representations of Polynomial Rings.

I got off on the paper with a good but not too good start — and it was really slow progress. By the end of the programme, I had managed to read and understand the proofs for only twenty of the eighty pages. Of course, that was my first experience, and there were many learnings I got from it which I feel will help me in future experiences (some of which are documented in earlier blog posts). Somehow, what I didn’t learn was to like deciphering a paper from start to end.

Another aspect of research and graduate life that frightens me as of now is the need to collaborate with others and work under people. So far, I have been working largely of my own initative, and most of my interaction with teachers is for learning specific subject material or clarifying doubts or general discussions. The notion of actually going through a paper with a guide, where I have to meet him/her on a regular basis and report progress, sort of scares me. The problem here is that it’s not just a commitment to myself or a personal choice, it’s based on and largely driven by the other person (particularly since as the guide, he/she is the senior and the one deciding the directions).

Of course, there are all kinds of guides and all possible arrangements one could have with the guide, and in fact, many research students have told me that the guides try to guide as little as possible and let the student decide the course.

Among other things that frighten me about my Ph.D. is that if I get too emotionally involved with trying to solve a particular problem, and if it doesn’t work out, then what do I do? And if I don’t come close to getting any new results within five years of getting started (or whatever the number of years is within which I need to complete my Ph.D.) How desparate will I get?

What happens if I have to abandon my favourite subject for a subject where I can get a result faster? What happens if the areas where I want to work turn out to be duds?

The other thing that’s bothering me about my Ph.D. is that it seems in some sense a final step — after that, I’ll stop being a student, I’ll be at the giving rather than the receiving end of knowledge (while doing a Ph.D. one is sort of at the interface between the two, one is trying to create knowledge while simultaneously attempting to acquire it).

So I want to utilize the time I spend in my Ph.D., to prepare myself as thoroughly as possible, for the road ahead. This is not just in terms of the mathematics content, but also the way mathematics and research is done, the various politics and administrative issues involved with research, the key to teaching and learning good mathematics, the various issues facing the mathematical and research community and the way they interface with other disciplines.

I’d ideally like to have a good network of people doing various parts of mathematics at various places, with whom I can interact and interface so that I can experience the feeling of doing mathematics in a community.

On the whole, it’s going to be an interesting and challenging phase, and the things I can do now (or perhaps over the next few months) are try to talk more to the people who are already doing it, and try to get over the mental block that I have (or think I have) with reading research papers.

I also need to wrap up, or reach a sort of logical point, with all the many ambitious projects that I have started and gotten underway right now. I don’t know how much time I’ll get to pursue these once I start out Graduate School, but I think there’s a fairly good chance it’s going to be far less than what I have right now. Which means I should go to Graduate School with these shelved aside (that doesn’t mean I won’t work on them — it just means that I won’t expect a regular output on them from my own side).

Let’s see how things unfold…

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