What Is Research?

December 21, 2006

Culture and diversity in mathematics

Filed under: Culture and society of research — vipulnaik @ 9:34 am

Recently, in India, there was great furore about the introduction of an additional quota for Other Backward Classes for reservations. These protests opened up a whole can of worms about why educational and professional institutions do not have proportional representation of different parts of the population. The pro-reservationist argument that if 20% of the populace of the country belongs to a certain social class, so must 20% of the populace of its educational institutions, rings true for its very simplicity.

Reservation actually reminds one of the closely related notion of Affirmative action. The aim of affirmative action is to positively discriminate towards groups or communities that have been discriminated against in the past. The obvious goal of affirmative action, of course, is justice, justice to communities and groups that have not had a voice so far. Another goal of affirmative action is discovering talents that would have been suppressed due to discrimination. The most interesting claim made by proponents of affirmative action, though, is that affirmative action promotes cultural diversity. The arguments for this are so often quoted that they sound trite: different cultures give rise to different perspectives, different ways of approaching a situation, and hence cultural diversity is important in any area. Hence, Proportional representation of all groups is the desired goal of any public institution, whether academic, professional, social or political.

If the argument sounds specious in this extreme form, we can still ask: to what extent is it true? How does cultural background influence the way one approaches a mathematical problem? Does being Asian, American, or African, endow one with particular ways of thinking that help in the solution of mathematical problems. In other words, how does generic culture influence mathematical culture?

First, what are the ways in which background can influence mathematical ability?

  • Genetic factors: Even though it may be politically incorrect to argue that genes could be responsible for mathematical superiority, the contribution of genes can definitely not be ruled out on a scientific basis.
  • Toddler education: The way a child perceives numbers and shapes is often shaped in his/her initial years. The quality and nature of mathematical education in these years can profoundly influence the child’s conception of numbers. Also, the practical situations in which he/she may use numbers at an early age can also play a profound role in the development of the child’s intuition.
  • Primary and secondary education: The nature of the curriculum, the course pressures, and the opportunity to express and develop talent in mathematics are definitely importnat determinants of how the child’s mathematical talents are shaped.

The argument for cultural diversity in mathematical institutions can gain strength if we show that different cultures are strong in these respects in different ways and to different degrees. That is, we need to argue something like: while a person from culture A is naturally better equipped for manipulating algebraic symbols, a person from culture B has a better feel for geometric figures. The important point, though, is that the arguments must actually be substantiated by hard facts. Let me look at a very clear and very definite cultural difference that is famous in the history of mathematics: Shrinivasa Ramanujan.

Shrinivas Ramanujan spent his childhood in Sarangapani Sannidhi Street, Kumbakonam, Tamil Nadu, born to a particular sub-caste of the Brahmins. Taking a simplified view, the Brahmins are the highest in a four-tier caste system in Hindu society, and the professed purpose of Brahmins is sacred learning. Thus, naturally, being from a Brahmin background meant that Ramanujan has the support and backing of his family (this is all in relative terms to the times) to play around with numbers. It also meant that others were willing to give him a more patient hearing and more of a chance to pursue and study mathematics than they would have given somebdy from a lower caste. In fact, it is inconceivable that a Ramanujan would have emerged from a non-Brahmin family at the time.

Thus, one cultural factor that played in Ramanujan’s favour was the fact that he was born a Brahmin.

What makes Ramanujan stand out is not just that he was a great mathematician, but that he had a whole lot of intuition and insight into mathematical truths, and could make astounding statements in mathematics even without formal proof. This has been partially explained by the mystic traditions in Hindu society. In Hindu society, the most rigourous of science developed alongside superstition and mythology, so much so that the two were completely intertwined and difficult to separate. The need for proof or rigour never arose; all results were justified by divine intervention. In this milieu, it was natural that Ramanujan tried to see the God directly rather than give conclusive and demonstrative proof.

This contrasts sharply with Europe, where the development of mathematics built on the foundations laid by the Greeks. Even though arguments by the Greeks were often flawed, they laid down the rules for logic and attempted to stick to those rules, very rarely seeking divine inspiration.

Thus, Ramanujan, as the intuitive mystic we know him to be, may not have existed had he been born in Europe. His talents may have led him to become a mathematician, even a first-rate mathematician, but the lure of the rigour of proof may have slowed down the steady stream of intuition which he poured out.

Is it?

It is really hard to say what Ramanujan could have achieved had he been born into a world where mathematical rigour was valued from Day One. It is possible that within this environment, he would have been able to combine his skills of grasping the results directly with the ability to give conclusive proofs, and gone much, much further. But there remains the possibility that he would have gotten himself restricted to a much more mediocre (though possibly happier) life.

Still, we do have one example of how a person’s culture is likely to have influenced his approach to mathematics.

However, the example of Ramanujan hardly gives grounds for the introduction of cultural diversity in institutions. Considering that the Indian culture could produce hardly any other mathematician, the fact that Ramanujan was produced by it seems scant justification for introducing a quota for the Indian perspective into mathematics.

In fact, what I feel is that mathematical culture is a very different ballgame from social culture. While social culture may have a few influences on mathematical culture, the fact is that mathematical freedom, creativity, expression and modes of thinking may often starkly contrast the level of freedom, creativity and expression and the modes of thinking existent in general society.

China, Japan, and states in the erstwhile Soviet Union, have produced a whole lot of mathematicians. So have countries like Hungary and Romania. What lies behind their success? A bit of genetics, a sound primary schooling system, and the development of a strong mathematical culture.

Hungary has a rich tradition of mathematics largely because it gave rise to mathematics greats like Paul Erdos, Bela Bollobas and so on. These people were not only a source of inspiration to others but also led to the creation of an atmosphere of mathematical contests in the country meant to sieve and train mathematical talent. The contests, the journals, the puzzles, all made people look forward to doing mathematics, they helped provide a meaning to mathematics.

A country like Russia, with its tremendous emphasis on a problem-solving culture, has naturally been producing great mathematicians. China, with its procedures for selecting and identifying talents at a young age and training them, has consistently been performing well at the Olympiads.

The United States has naturally benefitted and recognized talent and pushed it forward. But as such, the United States has very little of mathematical culture to speak of. It has no zeroes, no number systems, no pi’s and no right triangles to its credit. Yet, it has forged to be a leader in mathematics, with its educational institutions being the much-sought destination. Why? Because its educational institutions, at all levels, have sought to create their own mathematical culture. Each individual has brought in personal initiative and ideas towards fashioning a culture.

Arnold Ross kickstarted mathematics education for young kids in the United States, a precursor to United States’ current Olympiad programmes.

India is culturally poor in mathematics, because we Indians have been unable to create a culture for the kind of mathematics that is practised today. We cling to the importance of Vedic mathematics, spouting its cultural role, forgetting the fact that its primary role is only as a more effective means of computation — a goal that is itself outdated. We talk of the mystic element and the natural intuition of Indians in mathematics, with only Ramanujan to boast of, and he too being a moot point. Our ministers make misguided attempts to revive Indian culture by teaching us to treasure the zero and our numeral system and Vedic mathematics and linking all of them to mysticism and tradition.

And yet, the progress that India as a country has been seeing in creating mathematics and mathematicians is due to the mathematical culture created by individuals. We have not achieved cultural diversity, but at least we are on our way to getting a culture. Home Bhabha started off TIFR, a leading institute for mathematical research, where a culture for mathematical thought was developed. Alladi Sitaram, with others, started the Institute of Mathematical Sciences in Chennai, and this further fostered the mathematical culture that has been growing in South India since the time of Ramanujan. Seshadri started Chennai Mathematical Institute to propagate mathematical culture to students right from the undergraduate level.

Cultural heritage? Yes. But not the culture of tradition, of ancient values. It is not the invention of the zero 1200 years ago that moves a culture forward, it is the existence of a system for educating, harnessing and nurturing talent that counts. The past is just a stepping stone to the present.

If we seek cultural diversity within mathematical institutions, then the cultural diversity we need to seek is diversity of mathematical culture. The first prerequisite, though, for such diversity is that there should exist a mathematical culture in the first place. And to have this mathematical culture, there first of all needs to be an identification and training of mathematical talent.

What deep insights can an illiterate Indian give into mathematics by virtue of being born into a different culture? How does an African, who has no home support for the pursuit of mathematics and has struggled to study from textbooks publised in America, provide a unique cultural perspective to mathematics. How does a specialist hotelier bring new ideas to the study of differential geometry.

Cultural diversity in a community develops once culture gets developed. When the culture is well-set, trends start getting formulated. For instance, suppose a center for mathematics is established in a city, and young students come to that center to discuss and solve mathematical problems. Then, the cultural direction of that center may depend on who decides the collection of library books to be stocked in the center, who conducts classes and gives lectures there, and what policies the center follows. If the center decides to promote mathematical contests with a particular flavour of problems, a culture for those problems is likely to develop. If the center decides to recruit people and conduct lectures in a particular area, a culture for that area is likely to develop.

The mathematical culture itself derives from the mathematical institution rather than from any bare-bones culture. It isn’t Chennai that has a mathematics culture, but institutions in Chennai, be they Chennai Mathematical Institute or the Institute of Mathematical Sciences or the Association of Mathematics Teachers of India. It isn’t Cambridge, Massachussetts that has the mathematics culture but MIT and Harvard. A culture rarely comes imported from the past, the culture is created by individuals and institutions, and only then permeates to the city and the community.

Instead of seeking diversity for our mathematical institutions along the lines of social and economic diversity and diversity of generic culture, we need to seek diversity of mathematical culture. But, in the Indian situation, there hardly exists a mathematical culture in the first place. It is thus necessary to first create a mathematical culture, then let it diversify into many cultures, and then automatically reap the benefits of the diversity.

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!

December 7, 2006

It’s been a long time

Filed under: Regular updates — vipulnaik @ 11:07 am

It’s been a really long time since I last posted on the “What Is Research” blog.There are a variety of reasons. Firstly, of course, I was neck-deep (both in terms of time and in terms of mindshare) in a whole range of activities, including:

  • Admissions: I am applying to eight US universities and also appearing for the TIFR and NBHM examinations. Preparing applications, writing statements of purpose, preparing resumes, getting recommendations getting transcripts etc. have been taking a lot of my time! I have put up information about the places I am applying to at the Future plans section of my homepage.
  • Course work: This semester was light compared to previous semesters, but I still had six courses, four of which were credited. Of the other two, I did “Elementary Differential Geometry” almost as seriously as a credited course — I prepared hard for the Student Seminar and gave both the mid-semester and the end-semester examination. So the effort on coursework has also paid off.
  • Updating my webpage: I realized that for many of the things I want to share and write could best be done by putting up focussed material on my homepage, rather than blogging. So, I used the timeshare that would otherwise have gone into blogging in writing up about my academic life and areas of interest. Interestingly, a lot of this writing also helped me prepare my Statements of Purpose. I also plan to describe more about my non-academic life but that “plan” is way down my priority list at the moment.
  • A whole lot of volunteer work and other good work that absolves me of the responsibility of my time: Things such as devoting a little time to CMI Spark, a little time to Olympiad training (a one-day session at Mylapore), spending a whole lot of time chatting with friends, replying to emails, reading books. Things which give a lot of value but do not require me to be the creative proactive person.
  • Doing some real mathematics: It is interesting that this comes right at the end! That is not because it is the least important, rather it is because that’s what the rest of this blog entry is devoted to.

Before beginning the story of the real mathematics I have been up to, I’ll make a few observations in a similar vein to my previous posts. Firstly, my growing experience is making me realize that “working for oneself”, for one’s own goals, for the goals one truly cherishes, is the most difficult and the most challenging task. It is much much easier to work for others — to dash out a homework assignment, to go and sit inside class, to go and stand and teach a few poor kids how to read and write, to talk to other people, to worry about what others are doing or thinking. It is easy to “sacrifice” one’s time for others, whether in a noble or an ignoble spirit. Noble sacrifices may include “social work”, “being there for friends”, “sacrifices for the family”, and so on. Ignoble sacrifices may include sacrificing oneself to night of booze, sacrificing oneself by thetedium of procrastination, and so on.

The real challenge lies in finding out what one really wants and living towards that dream. Being too “busy” to pursue one’s heart’s desires is just as bad as being too “lazy” — they both boil down to the same thing.

So what is the true solution? Does the solution lie in not making sacrifices, in single-mindedly pursuing one’s goals to the exclusion of the needs and feelings of others? It is neither possible nor desirable to exclude the needs and feelings of others. Then how does one single-mindedly pursue one’s goals in the face of so many other noble and ignoble demands on one’s time and one’s mindshare? How does one find goals to cherish well enough to pursue in the first place?

I have discovered a partial solution, and I hope that works for me. This solution is recognizing the importance of placing my needs and dreams first, recognizing that my main purpose is to live up to my own ideals, and that being too busy to fulfil my dreams is just as bad as being too lazy The solution lies in recognizing that there’s nothing arrogant or evil about wanting to achieve something, about working to achieve something, even if the need to be “cool” or the need to be “good” demands something else.

The solution lies in learning that even if I do work for others, I am taking full responsibility for that work, I am taking full responsibility for the way it eats into my other time. I am not a helpless spectator seeing my time while away. At any rate, I should not be a helpless spectator.

These realizations have been hitting me gradually, over the last many years. Interestingly, these realizations have often helped me positively in terms of social activeness and awareness and all the “good things” — but that is not my aim.

Now on to describing my mathematical activities.

When I started off this semester, I was faced with many daunting tasks ahead. I had plans to apply abroad to a whole range of universities. This meant preparing for the General GRE (including a Verbal section and a Writing section both of which seemed to require lot of effort), the subject GRE (which would also probably extract at least some effort), and the TOEFL (which now had an all-too-tricky Speaking Section). It would also mean getting my “Statement of Purpose” right, getting recommendations from the right people, and getting all the other applicationnitty-gritties. I also wanted to take a number of courses. I had gotten myself booked for six courses already (of which two were audited).

So I knew life would be hectic.

But I also knew that just doing these things would not give me the deep sense of satisfaction I needed. There was one important unsolved mathematical problem and one important new idea that I had long hoped to work on once I was “free”. But I knew that freedom is an all-too-elusive concept, and so I decided that whatever time I get, whatever energy I have remaining with me after the application work and the course work, I will devote to these ideas.

The first couple of weeks came in getting the General GRE preparation under control, to the level where a few hours a week would do the trick. Once I had managed this, I wrapped up some documentation for my academic life and areas of interest, and then just plunged full force into the Extensible Automorphisms problem.

I had already done some work earlier on the problem, and discussed it with Professor Ramanan and with Professor I.M. Isaacs. I started off with my past ideas and results, and the past correspondence. There were already some ideas I had had earlier, which I had not yet tried properly. I began by formulating some of these lines of attack, and sent a mail on one of them to Professor Jon L.Alperin and to Professor I. M. Isaacs. Profesor Isaacs replied in the negative to my “conjecture”. His elegant proof immediately gave a whole new flavour to the problem and I continued to think and work on the problem to put the insights together. In addition, Istarted working on a whole number of parallel tracks within the problem, so much so that it became a mini-project.

The mini-project saw many ups and downs, and finally some time in November I decided to attempt a systematic documentation. By that time, however, things had become quite unmanageable. One of the problems was that many of the approaches required the introduction of new “local” terminology, and I was not sure how I could keep throwing in new terminology in a publicly accessible documentation. Then I hit upon the idea of a “wiki” to store the important definitions — the easy interlinkability within a wiki seemed ideal for what I was keen on.

Here, a lot of my past “good work” experience came in useful — last June-July, I had added many articles in Wikipedia, so I had a reasonable idea of the way wikis were to be edited. Also, as part of documenting CMI shifting concerns and also CMI Spark, I had come to learn of editthis.info. So I was all prepared to start off my wiki on extensible automorphisms.

Once I started the wiki, however, I realized that the kind of wiki I needed should have information on a whole lot of group and subgroup properties. This revived yet another old desire in me: a “property theory of groups and subgroups”. The winter holidays were approaching and I decided to plunge into creating an entire wiki on group properties. I’m still in the midst of that experiment, it is at groupprops wiki.

As I write this, I realize that once the groupprops wiki is well on track, I must return to my original task of documenting my progress on the “Extensible Automorphisms Problem”.

I’ll add a few remarks about my learnings from the Extensible Automorphisms experience:

  • Firstly, working on a problem all by oneself is really possible and exciting!
    Even if one has to squeeze in time for it.
  • It is one of the important axioms in research that one should work on problems that “others” consider important. This makes a lot of sense to me. But I have strongly come to feel that exploring new ways of organizing ideas and exploring one’s own problems can be of very special value. For one, it might lead to problems that are of importance to others. But more importantly, it gives a greater sense of personal control and of living up to what one really wants.

Parallelly with the Extensible Automorphisms Problem, I was also working on another “project”, the so-called “APS theory”. I have documented an entry describing this theory in my blog as well. This theory has, so far, not picked up pace. But I plan to bring it out and work on it soon.

The “APS theory” again began with innocuous observations I had made some time last March, and I had discussed these with Dr.Amritanshu Prasad
who had found them interesting. I had also discussed them with one of my Olympiad-time friends, Anand Deopurkar, who had also found them interesting. At the time, though, my ideas had been very raw, and I had worked on them during theTIFR Visiting Students’ Research Programme. Around September, parallelly with the extensible automorphisms problem, started working on this theory. Again, I followed a strategy of ruthless extensive documentation. But I never got around to explaining this theory to anybody, largely because I did not see how I could go and tell it to somebody (in the sense that I did not have a convincing answer to the question: what’s the use?).Around the beginning of November, I decided to shed work on the APS theory for my application process. Now, as I write this blog, I am wondering how to revive the APS theory. May be another wiki?

This theory taught me that building whole new theories out of thin air is dicey business, particularly when there is no external motivation to do so. But I still think it is an important and rewarding activity. It is enterprise. Every enterprising act, every gamble, does not pay off. But there are some which do. If my theory actually throws up something more than a mere formalism, if it throws up new insights into existing stuff, then it is a contribution to mathematics. More importantly, creating a theory gives me the theory-building flavour.

Writing this blog, I realize something that I probably knew earlier but never appreciated
so deeply. I realize that there’s a real whole lot of things I want to do (not all of which qualify for this blog, which after all is about “What Is Research?”) and a whole lot of things that I have a capacity for enjoying doing. This list is so large that at any moment, I am only working in the “top ten” of my huge list, and I consequently keep missing the things which aer not in the current top ten. So, going back to an activity that I’ve not managed to do for a lot of time (like blogging) makes me feel refreshed. It’s a bit like an old tune that I hear after a long time. The “hugeness” of this list also often makes me feel frustrated at times — particularly when an activity that is not in the current “top ten” is forcefully put on top of my consciousness.

Am I to consider myself unfortunate for this?

Definitely not. I am proud of my dreams, my desires and my ambitions. I am proud of the fact that I have too many things on my priority list. The one thing I have to watch out for is that the top items on my priority list are genuinely things that I want to do, that they are in my genuine priority list. I have no regrets about having “no spare time” provided that my main time is used in things that really fulfil my desires. The alternatives: having too little time because of being “busy” with activities that don’t matter, and having too much time with no worthwhile activity to spend it on, are definitely not attractive.

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