What Is Research?

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.

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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.

What others have to say…

Filed under: Uncategorized — vipulnaik @ 4:14 am

Indraneel gave me a nice link:

A Graduate School Survival Guide.

This piece is written by Ronald T. Azuma, a postdoctoral student in Computer Science who did his Ph.D. in 6.5 years. He tells us that while many of the lessons of graduate school are best learnt by experiencing it, there are some that one might as well know right at the outset. He details some of these lessons:

  • Be very clear about why you want a Ph.D. A Ph.D. is not lucrative on the surface: longer hours, delay in entering the workplace, frugal living, and many other demotivators. On the plus side, there is: the qualification and the preparation to do cutting edge research and expand the frontiers of knowledge. The author said he himself chose research because he wanted to contribute to knowledge, and he did not want that, five years hence, he would be in a job or position that did not satisfy him.
  • Understand that academics is a business and a full-time job: Academia is a peculiar type of business but a business nonetheless. The research guides and professors need to prove themselves to funders, and the students need to prove themselves to the professors. Resources are few and competition is intense.
  • Graduate school is about what you pick up and not what you are taught: Much of the learning in graduate school, especially for Ph.D., happens not in the formal courses, but outside the classroom, from books, from conferences, from discussions.
  • Many skills are needed: initiative, tenacity, flexibility, interpersonal skills, organizational skills and communication skills. The author details how each one is critical to performing well in graduate school.
  • Choose the advisor and committee carefully: The author lists advantages of choosing a non-tenured advisor: greater availability, greater personal involvement in the student’s research, a disposition for working hard (in order to get tenure). Advantages of a tenured advisor: greater experience, more resources and influence. The author balanced both by choosing a non-tenured advisor and a committee including some tenured persons. Other factors he says are important for choosing one’s advisor: (a) does the advisor push you to work? (b) is the advisor approachable? (c) is the advisor knowledgeable in all areas you want to work on?
  • Maintain balance and perspective: Getting the Ph.D. and churning out great research work is indeed top priority, but too much narrow focus on it can be damaging. A Ph.D. is like a marathon. Spurting unnecessary may tire one out early. The author alludes to Repetitive Stress Injury (RSI) as one of the possible consequences of an unbalanced focus on the Ph.D. goal.

A very well-written and illuminating piece. I think I find answers to many of my earlier questions in this piece, particularly my question on primary and secondary responsibilities.

The primary responsibilities of a researcher are indeed working on problems; helping others work on problems; reading, learning and attending seminars and conferences; and guiding younger students. But for a Ph.D. student, the relative priorities differ. From what I understand, the primary responsibilities of a Ph.D. student are more towards acquiring basic capabilities and
establishing credentials. So, the list for a Ph.D. student runs as:

(i) Reading, learning and attending seminars/conferences both in order to get a working knowledge of all fields and in order to decide the topic of study.
(ii) Interacting with people and building good connections to be able to choose a research advisor and committee
(iii) Developing skills and competencies related to working on specific problems that can realistically by completed within the framework of a Ph.D.

Thus, guiding younger students and working on improving the theory or working towards a magnum opus are not responsibilities of the typical Ph.D. student. The Ph.D. student should focus on demonstrating his/her potential by doing something in a short span of time that sets the stage for later magnum opa.

I have heard a few stories about how people with highly ambitious proejcts for their Ph.D. ended up taking 12 yars to do their Ph.D.

The research student also has seconday responsibilities, one of which is continually getting resources, time, money etc. towards primary responsibilies. This is extremely challenging, and Azuma discusses it quite a bit.

Hope you have a nice time reading Azuma’s piece!

Looking forward to your comments.

August 11, 2006

Primary versus secondary responsibilities

Filed under: Uncategorized — vipulnaik @ 10:20 am

In the previous posts, I discussed (or tried to discuss) how one goes about selecting a research problem. Having selected a research problem, one must work on it. “Working on a problem” has a specific flavour to it that I’ll come to in the next post. As warm-up, I’ll ponder about the kind of responsibilities a researcher has and how he/she seeks to fulfil them. My pondering here is largely uninformed, considering that I am still a B.Sc. student, and I welcome feedback from others on how things really are.

When I first entered CMI, I was quite astounded to see the extent to which mundane administrative and procedural responsibilities tended to consume the time and energy even for full-time researchers. When I saw people in other jobs working every day from early morning to late night, I wondered: what kind of a distracted life do people in the research world live, working only Mondays to Fridays, that too, often having to do a number of last-minute things that don’t directly pertain to their research responsibilities? Indeed, in my very first post, I had said: “Fun and inactivity accompany us wherever we go, but it’s the serious work we do in between that counts” to indicate that much of research is, at least apparently, inactivity (as far as research output is concerned). In this post, I plan to explore the reasons for this and what can possibly be done about it.

I begin with the distinction between primary responsibility, secondary responsibility, and others.

(i) A primary responsibility of a person in a role is a responsibility directly linked with the that role, such that the better the responsibility is fulfilled, the better. For instance, the primary responsibility of a person in his/her role as a parent is to support and take care of the child and help him/her grow. The primary responsibility of a student is to study and acquire mastery over the subject. The primary responsibility of a person as a waiter is to serve customers effectively.

The more effectively a person fulfils his/her primary responsibilities, the better. Thus, primary responsibilities are the areas where one needs to strive towards excellence.

(ii) A secondary responsibility of a person in a role is a responsibility that is needed or desirable for better fulfilment of primary responsibilities, but is not a goal in itself. For a parent, the primary responsibility of feeding the child may translate to the secondary responsibility of cooking a certain kind of meal everyday. For a person living in “clean” society, wearing clean clothes everyday is a primary responsibility, but washing them is a secondary responsibility. It is my primary responsibility is to get into a good university to study mathematics, it is my secondary responsibility to perform well on the GRE and TOEFL.

A person may exempt him/herself from a secondary responsibility by seeking another way to fulfil a primary responsibility. The parent may, instead of cooking the meal, decide to hire a cook. A person may, instead of washing his/her clothes, give them to the laundry or use a washing machine. I may on deliberation realize that preparing for the GRE and TOEFL is not so critical to admissions if I instead concentrate more on my grade point average.

Now, some questions:

(i) Do people often confuse between primary responsibilities and secondary responsibilities? How often do we take up relatively inessential tasks and treat them as primary responsibilities? How often do we ignore our most important and long-ranging objectives?

(ii) Does every job define a primary responsibility or is the primary responsibility determined more by the individual? For instance, do all mathematics researchers have the same primary responsibilities as researchers? Do all parents have the same primary responsibilities as parents? Do all school teachers have the same primary responsibilities as teachers?

(iii) What are the primary responsibilities of researchers, and more specifically, of mathematics researchers? What are the secondary responsibilities?

In this post, I try to explore (iii). Here are the usual responsibilities of a “mathematics researcher”:

(i) Working on (a) problems (b) theory building, both individually and collaboratively
(ii) Helping others work on their problems, by acting as a sounding board
(iii) Reading, learning and attending seminars in order to keep track of latest developments in the field and related fields
(iv) Guiding younger students at various levels

I believe that these are all “primary responsibilities” of the researcher, though the mix between them varies with the researcher’s seniority. For instance, a Ph.D. student may be expected to concentrate largely on (i)(a) and (iii) and possibly a bit on (ii) and (iv). A senior researcher may be expected to concentrate on (i)(b), (ii) and (iv). Nonetheless, everybody has some responsibilities of each kind.

Let’s examine more closely the life of a Ph.D. student. The primary responsibility of a research student is to work on his/her research problem, and also to learn/pick up skills. This may translate to any number of secondary responsibilities: meeting the advisor, preparing write-ups on portions of the research material, giving presentations to others, going for conferences, attending lectures, doing lecture assignments, discussing the problems on online fora.

A Ph.D. student affiliated to a college also has responsibilities to the college: responsibilities to participate in college events, to uphold the college name, to promote awareness of the college.

A Ph.D. student who is part of a regional community within a college has responsibilities towards that community as well.

My questions:

(i) What is the balance between primary and secondary responsibilities for the Ph.D. student?
(ii) Does the student wake up in the morning and say: “Hey! I’m moving closer towards my primary goals and fulfiling my primary responsibilities” or does he/she say every morning: “I have to do such-and-such for this secondary responsibility, such-and-such for this secondary responsibility….”

I’d like people actively doing research to post comments on what they perceive as their primary and secondary responsibilities, and how they manage to stay focussed on primary responsibilities while also taking out time for secondary responsibilities.

August 3, 2006

More on research and selecting problems…

Filed under: Thinking and research — vipulnaik @ 3:33 am
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Yesterday I talked to Indraneel, who is my senior ar CMI, and is now joining the Computer Science program at Princeton. I asked him how students select their research problems and work under advisors.

Some points that I learned:

(i) Students in U.S. universities are expected to finish their Ph.D. within five years. The first two years go in attending some courses.
(ii) The student must make a thesis presentation under an advisor in order to begin formally preparing for a Ph.D.
(iii) By and large, the student is expected to select his or her own resaerch topic. The advisor looks at the student and gives him/her feedback on whether the problem selected is a good one.
(iv) The student does not need to collaborate with the advisor on reearch. Indraneel said that many students he knew had written all their Ph.D. papers without an advisor.
(v) A student can finish his/her Ph.D. in any time within five years. If the student exceeds five years, then he/she must continue at his/her own expense, or the advisor must convince the university that the student’s delay has legitimate reasons.

Ph.D. is the first stage in life where a student selects his/her own problem and works hard on it. A very easy problem could be anticlimactic, while a very hard problem may result in a waste of several years. A problem in an area the researcher doesn’t like could mean five unpleasant years! There are two questions:

(i) How much choice does the student have in choosing his/her research problem?
(ii) How much choice does the student want in choosing his/her research problem?

The second question is interesting because a number of research scholars I talked to have said that they prefer that their advisor pick the problem.

This reminds me of the “love marriage” versus “arranged marriage” distinction. In a love marriage, two individuals marry due to mutual attraction and comfort with each other, whereas in an arrange marriage, the decision is taken partly by people other than those marrying. In a love marriage, the commitment is only to each other, whereas in an arranged marriage, the commitment and responsibility are also to the others who arranged the marriage. The main distinguishing feature of a love marriage, however, is that one must seek and confront one’s lover, fear rejection nad take full responsibility.

Analogously, selecting a research problem on one’s own requires the researcher to seek and confront, fear rejection, and take full responsibility. One cannot lean upon more “experienced” people.

Proponents of arranged marriage argue that arranged marriages have fewer expectations, are more based on the head than on the heart, and take into account the views of more experienced people. It is further said that even in an arranged marriage, love can develop after the marriage. On similar lines, proponents of “having the research problem chosen by others” argue that if the problem is picked by somebody else, they have less emotional attachment to it and can work on it more dispassionately. Further, a problem picked by an experienced advisor based on the student’s talents and preferences is likely to be more lovable than a problem the student would naively pick.

And one cannot fall in love on demand. In the same way, picking a research problem under external demand is difficult. On the other hand, it is easier and possibly more fruitful to work on a research problem assigned by somebody else.

Since the ultimate aim of selecting a research problem is to learn the art of loving and giving to the subject, I am (was) of the opinion that the student should select his/her own research problem. But how? How does one fall in love on demand? And is it really better or is my personal bias coming through?

Hope to have your comments.

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