My contrived success with the first paragraph of Kostant notwithstanding, I didn’t consider myself very successful with the paper. I wanted to get into the meat of things, and I hadn’t managed.
And for me, there has always been a huge list of other “to do” things. So when progress with the paper was lax, I started catching up with those other “to do”s, such as, updating my webpage, going out and meeting a friend, contributing meaningful articles to Wikipedia the free Encyclopaedia, and making myself useful to the community. Of course, with my lack of progress on the paper always at the back of my mind, gnawing at me. But guilt is something I have learnt to block out very effectively and the day swere so action packed that I hardly noticed how little time I was spending on the actual paper.
I wouldn’t say, however, that the time was a complete waste. You know how a tight can is opened, right? You try to lift it from one side, then loosen from the other and so on, till a critical amount is off, and then you just yank it. Now one way of doing it is to keep hard at lifting it and finish the whole activity in a single shot. The other is to try once, relax for some time, try again, and so on. Not so effective, but some work gets done anyhow.
One of the nice habits I developed from my early days in CMI was to keep documenting my observations. Actually the habit goes back to earlier when I used to prepare notes for the topics covered in school, though my notes at that time were understandable only by me. After joining CMI, I learnt how to write math stuff using the documentation tool LaTeX which you can download from here. This made it easy for me to typeset mathematical symbols and view a neat and clean version of my own creations and so I started using the computer more and more to document my knowledge.
Thus, even though I wasn’t relaly getting neck deep into Kostant’s paper, I kept recording my observations, and what I felt were the background motivations. My own ideas were probably at odds with Kostant’s, but at least it gave me the feeling that I was in control and doing something. Have a look at my initial findings right here. Sorry if that sounds like a muddle, that was what my mind said to start with! And also sorry if you’re wondering what Kostant’s paper really was, it isn’t legal for me to put it up online, so you’d better try getting it from a library.
Armed with this paper and a general feeling of inadequacy and uncertainty, I went and met DP.
Since I’d already sent DP my all too brief writeup in advance, I wondered: “Should I present what little I’ve written or ask him to explain some things to me?” I realized that although I had had a number of questions, I could not formulate them. Talk of getting tongue tied!
DP had already printed out my paper and marked errors. He seemed to have read the paper more carefully than I myself had. He had marked a number of errors. There was one thing he said he didn’t understand: the “Galois correspondence” that I had written about in my paper. I explained it to him.
I then explained to DP the procedure for computing the invariant subring of an algebra of functions from a space to a field, which brought algebraic geometry naturally into the picture. I used it to show DP that the invariant subring under the orthogonal group is the subring generated by the sum of squares polynomial.
I then confessed that I haven’t really made great progress. DP smiled at me and said that this result on the invariant subring under the orthogonal polynomials was an important (though easy) result and my coming up with the statement as well as the proof indicated that I had got the generic motivations correct. He then told me that invariant theory is an interesting subject and does not have too many prerequisites. Finally, he commended my habit of writing down all I was learning and told me to keep sending him updates.
The mathematics we discussed was very little. The main content of his advice was: “The cases yo uahve analyzed are the simple ones. The more interesting thing happens when the Lie group acts, not on the vector space, but on the Lie algebra by its adjoint action. There, we have a large nubmer of orbits of different shapes and sizes and understanding the orbits is a tough task. For instance, under the action of $GL_n(k)$, $k^n$ has only two orbits, but the Lie algebra which is $M(n)$, has a large number of orbits viz conjugacy classes.”
But it was inspiring and made me feel like working more on the subject. I realized the directions in which I need to explore. Discussions and guidance does help!!