What Is Research?

March 29, 2008

Frequently (?) asked questions

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

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

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

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

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

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

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

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

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

How does graduate life differ from undergraduate life?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The pros:

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

Cons:

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

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

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

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

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

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

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

How does the United States compare with Europe?

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

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

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

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

June 1, 2007

A skill set for research?

Filed under: Advice and information,Thinking and research — vipulnaik @ 9:15 pm

Looking back on my three years at CMI, and looking forward to five years of solid research in mathematics, I am currently trying to gauge: what is the skill set that I need to push myself through in mathematics, to see my ideas through (as I mentioned earlier), and to, in general, do well and survive?

Many of those things, I probably already have (though there’s always scope for improvement). For instance, I definitely have an ability to persevere with stuff, to concentrate for long hours. I also have a general propensity to document and organize ideas, to handle huge masses of information and ideas, and to come up with searching questions and search for answers to those.

Probably one of the things that I lack (not really lack, but could get more of) is the ability to do research alongside others, viz, to collaborate, to learn from senior people working in the area, to be able to work under another person’s guidance. I somehow feel (as I probably mentioned earlier in this blog) that I have too much of an independence trait, too much of a desire to do what I want, that may be it gets in the way of following what others say or advise.

Interestingly, I still do seek advice often; only I don’t usually implement it! I also often get started, half-heartedly, on a number of projects, and while I do tend to persevere on a few of them and see them to completion, I don’t usually close, or discard, the other projects.

Apart from being able to collaborate with people on an intimate basis, and take advice, i also need to get more on the social network of mathematicians. I think there’s quite a lot of sub-networks within mathematics students wherein the students share not just mathematics, but a lot of other things, and getting into these networks will help me feel part of a bigger community. Not having been able to do this much hasn’t really been my fault — on the other hand, I think I could have done better if I really wanted. It’s also something not unique to me.

I remember right from the time I was part of the Indian contingent to the International Mathematical Olympiad, that there were so many peopel from other nationalities whom I could have interacted with, but didn’t. It’s the same story out here, in France, four years from that time. Probably it has something to do with reverting to one’s familiar shell, doing the things one is typically used to (including both work and fun) rather than exploring what is new.

Another skill that I need to pick up is my presentation skill. As such, i am fairly good at giving mathematical talks, but presentation skills mean a lot more than just giving talks of mathematical content — it means presenting oneself to anybody in such a way that one can bring that person to one’s point of view! For instance, it may mean convincing a professor to back my research project, it may mean convincing somebody to sponsor me, it may mean a whole lot of other things!

Also I need to be a bit smarter about what the requirements at a place are, what it takes to convince the people of my sincerity as well as of my overall appropriateness. In CMI, I did suffer a bit initially because there were certain courses that I did not take seriously. At that time, I did not think that the knowledge I gain in them is of much use (which was probably true, but now I realize that coursework is an opportunity to gain a relationship with a teacher which could be useful later). I plan to be much smarter now. (In general, being smart about what counts and what doesn’t is an ingredient for success in any endeavour).

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.

August 29, 2006

What do graduate schools look for?

Filed under: Advice and information — vipulnaik @ 6:35 am

First, a little note about myself. I am in the third year of the B.Sc. (Hons) Mathematics and Computer Science course at Chennai Mathematical Institute. I am keen on doing further research in mathematics, though I am open to the possibility of switching to computer science. After completing my degree at CMI, I plan to join for an Integrated Ph.D. programme at a place that offers great resources and guidance. Hence, this year, I am applying to places. Given the current distribution of good research universities, most of the places I am keen on are in the United States, but there are a few in India.

Currently, I don’t have a very narrow area of interest/focus, and I enjoy a lot of the mathematics (as well as some complexity theory from computer science) that I have been exposed to. However, one area that has particularly caught my attention is group theory, and finite group theory in particular.

Universities that I am considering right now: University of Chicago, MIT, Princeton, Harvard , Caltech, University of Pennsylvania, and Wisconsin. Others such as Michigan, UIUC and Berkeley seem ruled out due to a four-year rule (they require four years of post-secondary college education). I’m not completely sure, though — I hope to confirm things further before ruling them out fully.

In my attempt to narrow down and decide which universities to apply to, I talked to Professor Ramanan, who works both at CMI and at the Institute of Mathematical Sciences. Professor Ramanan gave me the following pieces of advice:

  • On places to work in: There are two aspects. First, the place should accept me (be willing to take me in). Second, I should find an environment and faculty there to help me pursue my area of interst.
  • On how to look for an advisor: Professor Ramanan suggested I look for an advisor who has done a lot of good work, and is middle-aged (in mathematics, middle-aged means around fifty). Somebody too old may not be interested in taking new students and pushing them, and somebody too young may not have that much experience behind him/her.
  • On what the places look for while admitting students: The Graduate Schools are experienced — they look precisely for the qualities that make for a good researcher. These are the twins of focus and flexibility. Focus means the ability to zoom in on a specific topic and give one’s whole heart to it. Flexibility translates to the ability to change gears, change one’s area of focus, allow oneself to get interested in a new topic based on attending a seminar. Great mathematicians have done great work in one subject, then migrated to another subject based on a sudden fascination, and done great work in that subject.

Recently, I was reading the book How you can get richer.. quicker by M.R.Kopmeyer. He gives a very important piece of advice, that makes a lot of sesne for graduate school applications. Here’s the advice:

  • Figure out what they want, and give them more of it.
  • Figure out what they don’t want, and give them less of it.

To figure out what a Graduate School wants, it is enough to look at what they ask for. Below, I review the typical components of a graduate school application, and give my own opinions on what the School wants in each component.

Components that involve scores/grades/marks, where the direction of improvement is clear:

  1. GRE general scores: These are important to the Graduate School as they indicate basic verbal and quantitative abilities. The Graduate School wants students with a reasonable vocabulary (Verbal), reasonable passage comprehension skills (Reading Comprehension part of the Verbal), good quantitative skills (Quantitative), and the ability to analyze and express thoughts (Analytical Writing). It is also obvious why they want these skills — they are necessary for practically all academic work and social living. I believe that a score of 700+ (out of 800) in verbal, 800 (out of 800) in quantitative, and 4.5-5.5 (out of 6) in the essay/argument is good enough to give them what they want.
  2. GRE subject test scores: I haven’t investigated the Subject Test much, but its basic utility, so far as I can figure out, lies in its being the only objective way of evaluating the student’s aptitude in the subject. Its again not a big puzzle what the Graduate Schools want: people who are better at the subject. From what I’ve heard, I believe a percentile of 90+ is good enough to giev the graduate school what it wants.
  3. TOEFL scores: Listening, speaking, reading and writing are the four ways we send and receive information, and these are precisely what the new Internet-based TOEFL tests. While TOEFL scores in general are not so important, a good TOEFL score could be a plus point while applying for teaching assistantships. Basically, the Graduate Schools are looking for people who can handle the medium of instruction — English, with ease. Because the TOEFL pattern has changed, I’m pretty unclear of how much the Graduate School really wants.
  4. Grade point average: Performance in the undergraduate academic institution counts for a lot, particularly in the subjects that I intend to pursue for further study (in my case, Mathematics) and in secondary subjects related to it (in my case, Computer Science). Apart from using these to judge the student’s academic ability, the Graduate School also wants to learn, from the grades received by the student, his/her ability to survive in and cope with an academic evaluation system. Universities in the United States give GPA out of 4.0, and they expect the GPA to be around 3.7 or more, so that translates to above 9 on the 10 GPA scale used in CMI. You can check out my own academic record here.

In the coming points, we start moving away from what the Graduate School wants in terms of ability to what the Graduate School wants in terms of personality. Here are components that depend on past academic choices and skills demonstrated:

  1. Courses taken: The choice of courses that the student takes reflects the student’s interest and willingness to take up challenges. A senior told me that taking up a worthy course and getting a somewhat poorer grade counts for more than taking an easy course and sailing through with a good grade. Another senior told me that the entire pattern of courses a student takes determines the picture the student conveys to the Graduate School.
    Prima facie, it is unclear what kind of picture the Graduate School wants, or does not want. Whereas in the earlier four requirements (GRE general, GRE subject, TOEFL, grade point average), the direction of improvement was clear, here it is not. In fact, I don’t think there is any ideal pattern of courses for each student to take. Rather, the pattern of courses should fit in with other components of the student’s application, such as summer projects, extra achievements, and Statements of Purpose.
  2. Summer projects and summer schools: How the student spends his summer indicates not just the student’s talents but also his/her choices, priorities, and goals. A research life is a lot about choosing one’s topic and devoting time to it.
    Also, a student can, in the summer, hope to put in a much more focussed effort towards learning a topic, mastering a paper, or attempting an unsolved problem, that would not have been possible during term time. My seniors have told me that a person whose summer programmes are synchronized towards a clear and focussed goal stands a better chance of admission to any graduate school. The reason: graduate schools seek students with the ability to focus, and the best indicator of such ability is a past record of such focus.
  3. Extra achievements: While graduate schools do look for focus, they also look for a personality with wide-ranging interests. This means that extra achievements, extra activities will be viewed favourably by graduate schools if they indicate commitment to a cause, the ability to work hard, and the skills needed to succeed at arduous tasks.
    What graduate schols do not want is a string of impressive-sounding achievements in what they consider to be inconcsequential settings. Interestingly, the same achievement can be cast in terms that provoke very different reactions from the graduate school. The student needs to highlight what he/she put in and what he/she learned or gained. The graduate school seeks a person with certain qualities, and achievements are important only insofar as they highlight the necessary qualities.

Personal statements of the student and of others who know the student:

  1. Statement of purpose: Summer camps, extra prizes and honours, extracurricular activities, courses, course grades, are all just facts. The real personality comes through in the Statement of Purpose. This is not some test that the Graduate School subjects the student to in order to test his/her essay writing skills. Rather, it forces the student to clarify what he/she wants from the Graduate School and is willing to give the Graduate School. As I read in a book Wanna Study in the U.S., the Statement of Purpose should show the student’s life as a painting in the making, with the next stroke on the canvas being the student’s admission to and entry into the Graduate School.
  2. Letters of recommendation: This is an aspect of the application over which the student, apparently, has least control. Some recommenders prefer not to let the student see their recommendations, leaving the student in the dark as to how he/she is placed with respect to the possibility of admission. Like the Statement of Purpose, this component could make or break an application. I think what the Graduate School seeks from recommenders is confirmation of the student’s ability to fit in the bill for research life.

Here’s my overall feel: graduate schools are looking for people who are willing and capable to do research. Tests go only so far as to show ability, while personal choices show both willingness and ability. And the statements of the student and recommenders go further into showing both the willingness and the ability.

Research while an undergraduate?

Filed under: Advice and information — vipulnaik @ 5:43 am
Tags: ,

After finishing +2, I wanted to plunge into the world of mathematics. I had heard a lot about it: “it’ll be tough, there’s no money, it’s very abstruse…” Much of the superstitions turned out to be either false or irrelevant, but in any case, I was keen on putting up a good front, and coming up with creative ideas right from day one.

Now, it would be one thing not to get any ideas, and quite another to get an idea, work it out in great detail, and then not have any audience for it. But it is the second thing that happens in most research. Bulk of research goes incomplete. Bulk of completed research goes unrecognized. Bulk of recognized work goes unpublished. Even most of the published work is rarely ever read by anybody outside the clique.

I am reminded of the typical way a writer’s attempts at a story are described. Some of the greatest bestsellers have taken 20 rejection slips before getting accepted and published. An author took 640 rejection slips before his first acceptance, and then went on to write a plethora of novels and short stories.

Research doesn’t exactly correspond to writing, in the sense that there are more objective standards against which the worth of a piece of research or research publication can be measured. But it’s quite similar in one respect: experience, the ability to put things down, knowledge of the facts and the craft, count for a lot in taking an idea to something successful.

I’ll tell you a bit about the main ideas I’ve had, and why they are still far away from reaching somewhere recognizable. As I mentioned earlier on, I am more of a theory builder than problem-solver, my experiences with the Olympiads notwithstanding. Some of the big things I’ve tried to do:

  • A property theory that helps to study properties of groups, subgroups, languages and many other structures in a systematic fashion.
  • The extensible automorphisms problem which asks whether every extensible automorphism is inner. Check out my writeup on Extensible automorphisms (I plan to update this writeup in the near future). You can view a brief description at Unsolved problems.
  • A (currently underway) new approach to studying the matrix groups collectively, in terms of what I’ve called an APS.

Looking at all these, I can see the reason why I haven’t converted any of them to a recognizable piece of work. They are all involved wiwth reorganizing our existing understanding, and maybe furnishing simpler proofs of a few (already easy-to-prove) results. In summary, they are not importnat enough for other people to care about. What will really make one of these click is:

  • If I prove something that wasn’t proved. If I solve an open problem. For instance, if I am somehow able to completely settle the proof for the extensible automorphisms conjecture or obtain a more substantial result.
  • If I provide a substantially new proof for a hard result, a new proof whose utility is more than merely pedagogic.
  • If what I create provides a new perspective that is easy to adapt to and can be explained in a short time interval. If it knits together a whole lot of stuff. That is where I plan to take my work on APSes and my work on property theory.

Here are the attempts I have made to develop and convey my ideas:

  • I have created documentaries, as single files, as multiple files, with new terminology, old terminology, with examples, with everything. I have worked real hard on some of the theories.
  • I have approached some professors and lecturers with some of the ideas, usually picking on a person who may be somewhat interested in the subject matter, but more importantly, one who is approachable. Though they have always called the matter “uninteresting”, I don’t think it has genuinely excited them or that they consider it worthwhile to invest effort in.
  • I have written to people outside, trying to keep my writing and my work to a polite and decent minimum. Some of the people I wrote to replied. In particular, for the extensible automorphisms problem, where there was some correspondence with Dr. Martin Isaacs with whom I did some work towards settling the extensible automorphisms problem.

But I think this is only the beginning. Currently, I am really focussed on taking my APS theory some place. And I’ll do it… hopefully in a week’s time.

In the meantime, let me talk a little bit about a question that’s been bothering me for some time: how important is an undergraduate research and research publication in terms of giving credentials for further work? Here are some people who have got their research work published while still undergraduates:

Both the results were specific and important. Not earth-shattering or paradigm-creating, but important as steps towards an improved understanding. Tanmoy’s result was a step forward to understsnding the L-NL gap. Sucharit’s result went further in the study of free groups using topological methods.

Further, from what I can figure out, guidance from more experienced researchers who know the ropes is very crucial, specially for an undergraduate, to pick just the right level of difficulty in problem and the right approach. This kind of guidance, with the specific aim of solving open problems, is something I haven’t sought during my first two years. I have sought guidance for understanding of the subject matter and for understanding the motivations, but I haven’t made special efforts to ask for open problems and seek their solutions. Rather, I have been too closed working with my own open problems and pet theories.

Tanmoy got his idea based on summer work he did under Professor Meena Mahajan, even though that was in a somewhat different context (relating to the study of Nick’s class). He then worked out the details with Professor Samir Datta of CMI, and his result finally appeared in jount work with Samir Datta, Eric Allender and others.Another senior of mine, Indraneel, also got started in working on a problem based on discussions with a CMI alumnus, Raghav Ramesh Kulkarni, who is now in the University of Chicago Computer Science Department. They got a few results, and as far as I know, are trying to push their results a bit further before going in for publication.

What I’m probably driving towards is this. If you want to make some grand new big theory, like I did, then you’re probably not going to get it over while an undergraduate, because you lack the experience, the acumen and the time. On the other hand, if you want to work no a specific problem adn get concrete results, it’ just possible you can do it, but you need to have that dedication, that focus, and the willingness to be guided by people into helping you work on problems that they care for. Something that I didn’t do… and I think I should have. On the other hand, the only time I sort of did do it, was whne I was chasing up te extensible automorphisms problem, and I did manage to get concrete results.

I hope to be back soon with more detailed writeup on what I’ve been doing myself.

Hope to have your comments, and if you have any experience of doing undergraduate research, please let me know.

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