What Is Research?

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!

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December 21, 2006

Mail correspondence

Mathematics has often been accused of being a solitary profession, one that a person can practise without talking to anybody else, one that can be done in the head. One can keep one’s mathematical moorings completely to oneself. Like philosophy and realms of higher thought, mathematics can be carried out completely in the mind. Communicating the intricacies of mathematics is extremely difficult.

Paradoxically, though, the same factors that make mathematics solitary, also makes it one of the most social and communal of activities. The content and excitement of mathematics can be shared across several continents, through letters, through telephonic conversations, and of late, email correspondence. Mathematics as a profession allows networking oportunities for sharing of results and ideas that are not present in professions where physical contact and the “real world” are more important.

Sample the Hardy-Ramanujan story. Shrinivasa Ramanujan, a clerk in Madras, wrote a letter to Harold Hardy (of Trinity College, Cambridge) outlining some original results he had obtained in mathematics. His letter smacked at once of diffidence and self-assurance, his results spoke of great mathematical depth as well as lack of good mathematical schooling. Hardy went through Ramanujan’s letter, and saw the spark of genius in Ramanujan. Thus began a fruitful correspondence between the two, that eventually led to Ramanujan getting invited to Trinity College and working with Hardy on original problems.

Paul Erdos, the legendary mathematician, used to hop around the world everywhere, and yet he never lost touch with any of his friends. It was said that his typical letter began with: Let p be an odd prime…

Letters between mathematicians have often focussed not only on the exchange of mathematical content but even on general ideas in mathematics. The Grothendieck-Serre correspondence, for instance, has created new paths in mathematics at a time when the subject was undergoing a radical transformation.

Today, with the presence of instantaneous electronic mail, correspondence and communication in mathematics has assumed new levels of instantaneous. Imagine the kind of correspondence Hardy and Ramanujan carried out. Ramanujan sent Hardy a letter, it took a couple of weeks to reach (at least). Hardy then read it, wrote his reply, and sent it. That again took a couple of weeks to reach. The net result: Ramanujan had to wait for a month (at the very least) to get Hardy’s response to his results.

Today’s Ramanujan-equivalent can send the Hardy-equivalent an email in the daytime, and expect Hardy’s reply the next morning (by making use of the difference in day and night timings).

Email correspondence has provided us with a potent tool with which we can revolutionize mathematical communication? But are we using the tool effectively? Today, the equivalent of Ramanujan can try his/her luck with many a Hardy. But how many of us are willing to be brave and forthcoming, to overcome our diffidence, the way Ramanujan did?

The sense of community is very crucial to the development and fostering of mathematical research (or, for that matter, research in any area). Summer schools, workshops, seminars, are all aimed, among other things, at developing a sense of community and improving international networking. Today, however, we can build and enter communities through individual initiative, much more easily than before.

As an Indian, I say from some experience that Indians are naturally somewhat disadvantaged at building professional networking communities. The problem lies, to a large extent, with the general attitude of servility that has been ingrained into many an Indian through the social system, as well as the lack of practice in presenting and projecting oneself properly. On the other hand, none of these problems are unsurmountable.

Some questions I will look at:

  • What is the role and importance of email correspondence (with professors, faculty member and senior individuals) for a mathematics student, particularly at the undergraduate level?
  • What is the role and importance of email correspondence (with peers from different educational institutions) for a mathematics student, particularly at the undergraduate level?

With regard to the first point, it is true in the Indian context that the number of centers of excellence for mathematical education at the undergraduate level is very small, and even those that do exist are fairly small places as far as their mathematics department is concerned. Thus, many a mathematics student fails to find guidance in certain areas within his/her institute, and has only books, journals and the Internet to rely on. The student may be unable to pursue areas of his/her personal interest even in summer camps and research programmes, due to the inability to find a guide who specializes in those areas and is free to take the student on. Thus, the student may at many times be compelled to establish communication via email with somebody he/she cannot access more directly.

Another important incentive for establishing email correspondence is that it gives one a foothold in educational institutions where one may later seek admission for study or summer programmes. For instance, after completing my undergraduate studies, I plan to apply for Ph.D. in mathematics to various places in India and in the United States. Having corresponded with professors in some of the universities I am keen on, I feel a greater sense of confidence if what is going on in the institution and what I can expect once I join.

Establishing email correspondence is also good practice for joint work. My email correspondence with Professor Martin Isaacs of the University of Wisconsin-Madison led to a partial solution of the Extensible Automorphisms Problem and also helped me get a better feel of representations and characters. Further, it have me insight into how one usually goes about solving new problems.

Email correspondence can also increase general awareness about certain areas of the subject that are neglected in one’s own institute. It gives the cross-cultual facor. I got important pointers on where to read up groups and subgroups, as well as some subtleties in the subject, through correspondence with Professor Tuval Foguel of Auburn University-, Professor Derek J.S. Robinson, and Professor Jonathan L. Alperin.

Regarding the usefulness of email correspondence with one’s peers in other institutes.

The advantages are quite similar: there is a natural cross-cultural factor, one stays in touch with the way education is proceeding in other institutes. A student studying at another institute may tell one about interesting courses at that institute, and thus help create a new area of interest. Such a student may also be a valuable source to connect to other senior people at the institute.

I haven’t maintained a large amount of correspondence with students in other institutes (perhaps unfortunate). I have had sporadic contact with my Olympiad-time colleague Anand R. Deopurkar, and of late I have also been staying in touch with some people one year senior to me, who are at various Graduate Schools. Just talking to them and knowing the situations in their various schools has been valuable input for me.

The important thing about initiating and managing one’s own email correspondence, though, is not just what it achieves, but what it symbolizes: individual initative taken in the direction one wants to proceed. Rather than limiting oneself to the resources offered by one’s own institute, one actively takes one’s fate in one’s own hands and proceeds to aggressively fulfil one’s own interests.

So how exactly does one go about establishing email correspondence? What are the pitfalls?

I am pretty much a novice in the area, so my observations are still in the process of getting collated.

  • Write to a specific person for a specific purpose. There isn’t much point writing to a person just because he/she has won a Fields Medal. Communication with a person should not be done based on the person’s stature, but rather based on what one seeks to get from that person and whether that person is well-equipped to help in that direction.
    I have noticed that many people seem to think of writing to outside people as a matter of raising one’s personal prestige, a bit like moving in exalted iintellectual circles. I think this is an inappropriate attitude because it has implicit assumptions of academic stature taking precedence over the utility of correspondence. It is probably a legacy from the era when knowing the high-ups in an intellectual endeavour is what counted for success.
  • Give a brief description of why you are writing to that individual person. For instance, if writing to a person on a knotty problem in string theory, you can mention (truthfully) that you have come across this person’s papers or personal webpage in the subject, or that you have heard of his/her work in a course or from some other individual.
    This is not meant as an opportunity to give a glowing testimonial to a person whom you probably don’t even know. Glowing praise for a person you don’t know sounds like fawning servitude.
  • Give a brief description of the problem and make it very clear what kind of input is sought. Looking at the many attempts I have made at correspondence, the following stands out: in cases where I set forth 1-2 very clear questions and described the problem accurately, the probability of response was much higher.
    Often, students who have a whole lot of their own ideas and have not had the opportunity to discuss these ideas with anybody around them or close to them, seek to make full use of email correspondence by waxing eloquent on their ideas. This is usually couterproductive. The average person does not want to hear your new ideas up front. Present him/her with your questions first, let him/her respond, and then follow up by disclosing your ideas. If it is necessary to first describe your idea in order to ask a question, give a small and self-contained description.
  • Ask the other person to point you to references for further study and areas where the problem has been previously considered. By saying this, you acknowledge that it is possible that the questions you are asking may already have been answered somewhere, and that you seek guidance in locating the answer. This also shows to the other person that you are motivated to study yourself and are not using him/her as a doubt clearance service.
  • If it fits, give a brief explanation of why you were unable to resolve the problem from standard references, and are eager for further guidance.

What happens after the first mail is sent?

If you don’t get a response, do not be disheartened. There could be a lot of reasons:

  • The person was on holiday, or on a conference, or travelling, and is not checking mail.
  • The person no longer maintains that email address.
  • The person missed out your mail.
  • The person did not find your mail of much relevance to his/her area of interest and hence forgot about it.
  • The person read your mail and will take time out to reply after a few days. While many people respond in a day, it usually takes about 3-4 days.
  • The person is mulling over the contents of the mail.

All these are much more likely than what people often conclude:

  • This person is too high to answer a lowly creature like me.
  • May be that mail was so stupid that the person didn’t even read it.
  • May be i shouldn’t disturb people with such silly ideas and questions.

The advisable course of action in case a person does not respond is to just leave it at that. Of course, investigate the content of your mail, see if you have made any mistakes, and try to find out if the person usually responds to mails. It is best not to send a reminder or follow-up mail, because that sounds like you are holdign the other person accountable and accusing him/her. However, you can send him/her another mail after some time on a different or related topic. Do not try to infer conclusions about the other person being too busy to have read your previous mail. Best not to mention it at all, except perhaps as way of introduction (I had written to you earlier on…)

Once you do receive a reply, go through the reply carefully, mull it over, and send the next mail after you have either done a further round of processing on the reply or with a different doubt. Remember in the next mail to acknowledge previous correspondence (by way of introduction) but not make a big show of it. The worst mistake is to expect the other person to still have your previous mails in his/her inbox. Make each piece of correspondence completely self-contained, making no demands on that person’s memory of previous correspondence.

Remember also to keep track of all email correspondence with each person so far.

Email correspondence is a really fruitful way of expanding one’s mathematical boundaries and working for one’s mathematical future. It’s definitely been that way for me!

October 13, 2006

Reading research papers

From what I’ve gathered through talking to people on their way to a Ph.D., there’s quite a difference between the mentality required for coursework and the mentality required for research. For instance, in a course, the subject matter has been distilled and organized in a particular manner by the instructor. There is a clear path to follow: attend the lectures, read the lecture notes, read the text books and other references, solve problems, do the assignments, and sit the examinations. Even if everybody does not follow this clear path, the fact that it exists is a source of reassurance.. it is always there to fall back upon.

Doing research, which may involve solving open problems or extending existing results, is a different ballgame. At best, the student is given a problem and some material to chew upon and is then practically let loose on it. At worst, the student is told to find his or her own problem and work on it and keep using the advisor for course correction. Clearly, a completely different approach is required for this: an approach where the student figures out what information to collect, how to collect it, how to use it, whether to discard it, and so on.

An important distinguishing feature about research orientation, then, is broad reading with a narrow focus and a specific objective. Since this kind of focus cannot be provided in the routine college environment, students keen on developing the research orientation need to find other means of developing the skills. Summer schools and summer camps usually help in providing such focus. For instance, in the VSRP programme at TIFR, that I attended this summer, I was asked to read a paper on Lie Group Representations of Polynomial Rings. Here’s a link to the final presentation I gave on a part of the paper. I’ve chronicled about this paper in earlier posts on this same blog. Check out this post and subsequent posts.

Students can be encouraged to read papers even within coursework, by having student seminars as part of the course accreditation. Some of my courses at CMI this semester have student seminars. For instance, in the course on Representation Theory of Finite Groups, each student is supposed t give a seminar on a topic decided by the instructor; I have to give my seminar on Artin’s Theorem. In the Elementary Differential Geometry course (course details are available here), a list of seminar topics was given and each student had to select a topic, I chose the Whitney embedding theorem and a write-up of what I presented is available here.

Apart from reading research papers for these courses, I have also been reading research papers to seek and collect knowledge in various areas of mathematics.

First, some differences between the textbook and the research paper:

  1. The textbook presentation, or the lecture note, is meant to be an introduction to the subject. It is intended to provide overall motivations, basic definitions, and a level of familiarity and comfort to people who are new to the subject. Steps are left out or missed only if they are easy for the reader to fill out or filling them is an instructive exercise for the reader.
    The research paper, on the other hand, is meant to be a concise introduction to a new discovery or a new idea or a new formulation, for people who are already familiar with the area. Definitions and background are provided only in order to set notations and conventions, explain the authors’ mindset and revive the memory of readers. Efforts are not made to be complete. Further, the authors tend to skip on steps which: (a) have been proved elsewhere (b) require routine checking that other experts can do (c) provide no insights and detract from the essence of the paper.
  2. A (well-written) research paper has a clear end in mind, which it tries to outline in the beginning. It then gradually builds up the arsenal and ammunition needed towards proving this end. At some point in the paper, the authors usually discuss how this new result sheds new light in the areas being explored.
    A textbook, on the other hand, may not have a clear, specific result that it intends to establish. Rather, it aims to develop a backdrop and a framework in the minds of students.

Based on my experiences (both positive and negative) in trying to grasp research papers, I have come up with the following strategy:

  1. Try to get an idea of what the paper is trying to prove. This can usually be gleaned from the abstract, from the introduction, or from the beginning of the second section (if the first section is for preliminaries).
    Look for something marked Theorem 1 or Main Theorem.
  2. Understand carefully the statements of previously written results in that area, and use that understanding to try to figure the import of the new result obtained. Try to state the new result obtained in as many different flavours as possible. Make all of them as appetizing as can be!
  3. Now, look at the statements of the lemmas and corollaries, and try to understand each statement. Attempt a broad trajectory that describes how the theorem is obtained, via the lemmas and corollaries. Do not look at the proofs yet, unless they help significantly in understanding the statements.
    While trying to understand the statements of the lemmas and corollaries, it may be necessary to familiarize oneself with the notation of the paper.
  4. After a short break, look at this trajectory, and try to figure out which steps in the deduction process are clear and obvious. Often it may happen that many steps in the deduction process are not too hard. Figuring out that one already understands a lot of the proof before having seen the actual proof is a great confidence-booster.
    For the parts where the proof seems clear, look at the actual proofs and see whether they match the proof in your mind.
  5. Now, it is time to focus on the non-obvious parts of the proof. Gently look at the proofs of each of these. Some of these may turn out to be clear once you read the proof. For others, however, the proof may involve some new idea. Zero in on the proofs that are hard to understand. Note the crucial leaps of thought. Don’t be in a hurry to digest these pieces.
  6. Come back after another break. Recall the proof skeleton, and the proofs of the easy part. Now, in easy sessions, master the hard parts. Take special care to master those parts that fill you with the maximum discomfort.

This approach steadily zooms in on the proof details by beginning at the main result, then proceeding to the proof skeleton, and then finally going to the nitty-gritties of the actual proof.

What are the kind of results one obtains with this approach?

Some observations:

  • Steady documentation at each step is particularly useful. In this respect, I think one way of documentation is to prepare a presentation on the paper. A nice tool for preparing presentations is the document class beamer in LaTeX.
    Here’s an example of the PDFized version of a file using beamer: An electric story of a drunkard. The original LaTeX file looks like this.
  • Often, reading a research paper is disconcerting because one realizes the many gaps in one’s knowledge on encountering statements that the authors claim are obvious but that are not obvious at all. This has happened to me quite often! But whenever I have followed this zoom-in strategy of first concentrating on the broad motivations, then strengthening the proof skeleton, and then going in for the actual proof details, I have found that the disconcerting parts only come towards the end, by which time I have already gained a lot of confidence in the paper.
  • The “zoom-in approach” works best if the reader is used to looking at things and ideas in terms of their motivations, and understands the broad motivations in the topic where the research paper was written. These motivations are meant to be developed in the regular coursework, through comments and remarks made by the instructor, through the structure of the course outline, trough comments in the book, through the choice of exercises and problems that the student solves.
    However, even students not used to looking at things motivationally can start doing so by applying the zoom-in approach to a given paper!

Now, a chronicle, of some of the mistakes I have made when reading research papers:

  • Reading the first two pages and then quitting: True, this isn’t a really bad thing if the paper is well-written, because the author would have put the statement of the main theorems in the first two pages. However, simply knowing the statement of the theorem, without understanding the proof skeleton, may sometimes be useless.
    In some cases, the proofs may be hard. But in the past, I have often skipped the proofs simply because they seemed too tedious. However, now that I have started applying the “zoom-in” approach, I am able to absorb a little of the proof skeleton even if the steps of the actual proof remain unclear.
  • Getting disheartened because many statements in the beginning don’t seem to make sense: The introduction of the research paper usually contains both background preliminaries and a summary of important results shown in the paper. While reading the paper on Lie Group Representations of Polynomial Rings, I thought that the first few pages contained background preliminaries, and was disheartened at the fact that figuring out their meaning took me a lot of time. Only after crossing those initial pages did I discover that the content of the first few pages was not background preliminaries, but results proved in the paper.
    To avoid confusing background preliminaries (viz what is assumed) and the core content of the paper (viz what is established/proved) it is important to have a look at the whole paper. A strategy that I have followed since the experience with the Lie Group Representations paper is to create a mapping of the introductory section onto the rest of the paper. This way, it is clear to me which parts of the introduction have what purpose.
  • Not having any clear targets: A huge research paper can be daunting, but at the same time, it may be difficult to set intermediate targets. That’s what happened with the Lie Group Representations of polynomial Rings paper. It took me a lot of time to get a hang of the structure of the paper.
    In retrospect, I feel that after mapping the paper, and getting a hang of its structure, I should have singled out the results that it was important for me to master, and then applied the “zoom-in” approach towards mastering them.

I’ll post more on this. Looking forward to comments in the meantime.

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