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 Bulk of recognized work goes unpublished.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:
- Sucharit Sarkar published a paper on Commutators and Squares in Free Groups while doing his B. Math from ISI Bangalore. Sucharit is currently working towards a doctoral degree in the Princeton Mathematics Department.
- Tanmoy Chakraborty obtained the result that Reachability for Single Source Planar Directed Acyclic Graphs is in Logspace and had it published in a joint paper. Links to papers and presentations are available at Tanmoy’s homepage. Tanmoy has just finished his B.Sc. (Hons) in Mathematics and Computer Science from Chennai Mathematical Institute (in fact, he’s one year senior to me at CMI) and he’ll now join the University of Pennsylvania.
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.