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.