In an earlier post on this blog, I had outlined a so-called APS theory, which had roughly the following highlight: we take a sequence of sets and define maps from the set corresponding to m times the set corresponding to n, to the set corresponding to m + n. Sometime in the second week of May, I had discussions with a student Olivier Dudas, and with my advisor Olivier Schiffmann, where I learnt some interesting things about the general linear groups and orthogonal groups. These tidbits that I picked up led me to come back to the APS theory. I looked at it again and found that many of the ideas in this theory could be developed further by me now that I know a bit better about things like Dynkin diagrams, root systems, Chevalley groups, Lie algebras, and the peculiar relations between the general linear group and orthogonal group.

Towards this end, I focussed on exploring more closely the rough sequences-theoretic notion of algebraic extension that I had developed earlier. Roughly the idea is that just like we say that a field is an algebraic extension of another field, in the same way, we should be able to talk of algebraic extensions of APSes, that generalizes that of fields. In essence:

- If L is an algebraic extension of K of degree d, then the APS of general linear groups over L is an algebraic extension of degree d over the APS of general linear groups over K.
- The general linear APS over any field, is an extension of degree 2 over the orthogonal APS over the same field.
- The unitary APS over the complex numbers is an extension of degree 2 over the orthogonal APS over the real numbers.
- If we have an algebraic extension of APSes of algebraic groups, then this also gives an extension at the level of their Weyl groups. Thus, the fact that the general linear group is an extension of degree 2 over the orthogonal group, translates to the fact that the symmetric group APS is an extension of degree 2 over the signed permutation APS.

Equipeed with this and many other related areas, I went and talked to Professor Schiffmann. I also asked him whether this APS-theoretic study can be extended to other related structures and constructs such as the associated Hecke algebra. Professor Schiffmann was quite enthusiastic and interested, and suggested that I look at the correlation between this and invariant theory.

In particular, he focussed on the part about how the general linear APS over complex numbers was a degree 2 extension over the general linear group over the real numbers, and suggested that this could have to do with the theory of invariant polynomials. he even gave me some concrete links. Unfortunately I haven’t been working on the theory, or on invariant theory, much since I last met him!

However, there were a few other things about the meeting with Professor Schiffmann that impressed me.

One was the fact that even though he was quite enthusiastic, he was very clear that “my” theory was my responsibility and my work and it was upto me to decide whether to spend (waste) my time with this theory! That is, it was up to me to prove the worth of the theory, the world was not going to take it up and say — “this is interesting, let’s develop it further”. This was clear from the way he always emphasized “You could look at this further in relation to your theory”.

Also, his emphasis was that if I want to give credibility to my theory, then I should forge strong relations between that and existing important results and paradigms in mathematics. This is something that I have felt myself, but his repeated emphasis on whatever interesting results and framework are already known, helped reinforce the fact that to be accepted into a body of knowledge, a theory must first integrate itself completely into that body of knowledge!

The above things may seem obvious in hindsight. On the other hand, there was a time when I was very young, when I had dreams of doing some funny thing like writing great stuff that everybody would want to read! Basically the idea that whatever I said, people would puzzle over and try earnestly to interpret and follow.

But I have realized that such things don’t happen. Life (and even research life) is not just about doing great things, it is also about advertising those great things, it is about putting those great things in a framework that others can follow, that others find easy to accept. In other words, it is about conforming to a system into which one wants to be accepted.

Tihs learning has come to me, in its true form, very gradually — and I am probably still learning it. In my first year at CMI, I thought — okay, if I have a great idea, then that’s it! But then I realized that a lot of effort goes into packaging an idea, into conveying it to the right people, into making it ingnite into something that is actually useful. I also realized that even a bad idea can be implemented much better and thus be of much greater use, than a good idea which one doesn’t bother to implement and work hard at. In fact, much of research, from what I gather, is fairly average ideas that people have bothered to implement to the point of completion. The great ideas may be sources of inspiration — which few but the best can implement!

This brings one to the interesting notion of *seeing an idea through*. This is the commitment to having faith in a particular idea (which may be a *goal*, a *conjecture*, a *method*, whatever) and doing whatever it takes to make that idea reach a stage where it is accepted by a larger community as being useful. There are, of course, lots of problems with this. Firstly, how does one judge beforehand whether an idea is worth seeing through? Secondly, why is it important to see an idea through?

The first question is important and interesting because most of us don’t have ideas, and even those who do have, rarely want to see them through (I do have ideas, but at this stage, am not sure whether I want to see them through). Rather, as doctoral students, we often may ask our advisors for the ideas and problems, which we then try to see through. Advisors in general may be better able to tell us what we is (a) more likely for us to be able to see through (b) more useful to the community if we see through.

However, I feel, and still believe, that if a person is able to come up himself or herself with something that he or she can see through to the very completion and which in itself has some impact (however small) on the way a community thinks, then that achievement counts for a lot more than following an advisor’s direction on picking what one wants to see through. Of course, it also has a lot of inherent dangers, since how can an inexperienced student know what is important, and how that can be attained? But that really *is* the kind of thing that *I personally* want to do — find something that I am truly passionate about, something I have discovered myself, and see it through till the very end.

Of course, I am not sure that the things I have come up with right now (or am toying with right now) are anywhere near the kind of things I can see through (though I think APS theory holds a good chance, but let’s not be too premature’ I have a lot to see and understand). I am also not sure about whether the people I get as doctoral advisors will have some agenda for me; on the whole, I’d rather get people who have some broad agenda, than people who just let me do what I please! But what I hope for is to find a setup wherein what I personally want to see through matches with the agenda that is set for me (upto a little adjustment on my side).

But I do feel I have what it takes to see things through. It is not clear where this confidence arises from. Probably it comes from the fact that generally, whatever I decide upon, I get done, even if not exactly in the way I had planned! Also, my efforts with the Extensible Automorphisms Problem, with my Group properties wiki (group theory wiki), and whatever I have done towards APS theory, have greatly incresed my confidence and my ability with processing and organizing information and solving particular problems.

What I want to explore right now, at the ENS, and in general, is: what is the skill set that I need to develop to be able to see stuff through? To be able to persevere at stuff, build stuff where it didn’t exist, to be able to convert an idea into something that makes an impact on a community? I will explore answers to these questions in the coming blog posts.

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