Today, the VSRP programme at TIFR concluded. My project was to read and understand a paper on “Lie Group Representations of Polynomial Rings” and I have mastered the first section of the paper. My guide was Professor Dipendra Prasad, who was also the VSRP coordinator. For brevity, I shall refer to him as DP.

In the last two days (Thursday the 13th and Friday the 14th) of the programme, each VSRP student gave a Concluding Presentation from material learnt during the programme. The audience for the Concluding Presentations included other VSRP (Mathematics) students, Professor Dipendra Prasad, Professor M.S. Raghunathan, Dr. Riddhi Shah, and other interested faculty and research scholars. The venue was room AG77.

The lecture schedule was tight. Nine speakers had to be accommodated into limited time: two there hour slots minus tea breaks. So, all in all, each speaker could be allotted only 30-40 minutes. That is a very short time to present what was learnt in a whole month.

On Monday itself, I finalized my Concluding Presentation topic with DP: “Module theoretic freeness over the invariant subring”. I had already written a document on this and some related parts of Kostant’s paper which can be found here. Moreover, this was an “in itself” subtopic of the paper and I felt that I could do reasonable justice to it within the short time provided. The question was: how should I present?

A blackboard presentation seemed the obvious choice. AG77 has six blackboards, or, more precisely, a sextuple blackboard. (Check the P.S. for more). So blackboard space was no constraint. Experienced TIFR lecturers used the board admirably and with dramatic effect.

However, time for the talk was limited, and from my own experience, writing on the board is tedious and time consuming. During the recent Microsoft Research Summer School, I enjoyed attending computer based presentations, some using LaTeX (converted to PDF) and some using Microsoft Powerpoint. I wondered: why not try making a LaTeX presentation for my Concluding Presentation?

Some mistakes/problems that I wanted to guard myself against:

(i) Too much time writing stuff on the board: Writing and speaking at the same time becomes a strain.

(ii) Multiplexing the board use: The boards need to be used for stating results, putting down notation, giving proofs and a number of other things. There are two ways of using the boards. The naive way is to just use the board like reams of paper and fill one board after the other. The more sophisticated way is to reserve some boards for important stuff and use other boards for scribbling and side stuff. Unfortunately, the clever way requires planning and careful thought and a lot of experience, which I didn’t have.

I wanted to use the board cleverly, but that would be possible only if I limited and clearly specified my use of the board. Then I could decide what to keep, what to erase and so on. I didn’t want to get trapped multiplexing the board incorrectly.

(iii) Not concluding properly: The time constraint was an important factor and almost all presentations overshot the time limit.

So I wanted to make sure that I conclude on the right note rather than just peter off with nothing more to say. Thus, I decided to prepare some breakpoints in between.

(iv) Losing the audience right at the beginning: This is something that happens in many mathematical talks, and I wanted to avoid it. I wanted to let people decide whether or not to listen to my talk only after they at least got to know what my talk was about.

(v) Letting people think they can correct me whenever they want: In a short and highly time bound presentation, the most dangerous thing that can happen to a speaker is getting sidetracked by people pointing out errors, making comments and suggestions, and asking for more detailed explanations. I determined that I would strive to avoid this.

With these points in mind, I began preparing my slides. Though I had had a lot of experience with LaTeX, and I had tried my hand at writing slides in LaTeX as well, this was the first strongly mathematical slide show that I actually planned to present. I first made a preliminary version of the file and printed a paper version. This paper version began with the Prerequisites, the Nice to Knows, and the Goals. Each piece of matter was classified as Setup, Goal, Question, Proof Idea, Ingredient, Exploration and so on.

I then started finding out how many of the prerequisites the audience knew. Since many of them were not aware and not comfortable with some basic terminology, I thought I must spend time defining it. But when I did a test trial of things, I discovered that the test trial spent ten minutes on just developing prerequisites. So I decided to skip detailed explanations of the prerequisites.

After this “trial” I decided to add two footers to each slide: (i) Expected time, and (ii) Blackboard use. This was meant to be a guide both to me and to the audience. In case somebody in the audience asked a time consuming question, I would just indicate the expected time for the slide, give a quick answer, and move on.

I also made a mental plan of board use. I would use one board column to store important notation, another board column to store important terms defined, and another column to store results that I have stated but for which I have not provided complete proofs.

Thus prepared, I gave my talk. Here are the slides.

There were no interruptions (except one asking for the slide to be repositioned and one asking a trivial question about term ambiguity). So I just went on and on. 35 minutes into the show, DP asked me to start wrapping up, and I replied saying that I was on the penultimate slide. I finished within less than 45 minutes, overshooting my time limit by only 5 minutes.

DP commented that while my talk presented the main proof ideas, it did not give examples, although I myself had, in my discussions with him, viewed the examples in great detail. He further remarked that a thrust of Kostant’s paper was: how to work through specific examples?

I replied saying that my presentation was focussed on a small segment of the paper. I considered the proof ideas to be the relatively hard part and viewed this talk as a way to get an interested person over the hurdle and get straight into the paper. While preparing the slides, I had not considered working out example cases.

In retrospect, I realize that working out some special cases would have given the newcomers to the subject a better flavour of things. I’ll keep this in mind for my next presentation.

If you have comments on the slides, or on anything about presenting slides, do post them here.

P.S.:The “blackboard area” has three vertically separated parts. In each part, there are two sliding blackboards — the front and the back blackboard. Moving one of these up moves the other one down. (Check out a sliding blackboard arrangement to get an idea.)