Part III Seminars - just do it.

Why? 

• At some point in life (unless you are seriously unlucky) you are going to stop being a student. Hard though it may be at this point in Part III to imagine it, there is an end to sitting in lectures, taking notes, doing examples sheets and dreading exams. Whatever else is going to come After, you will almost certainly need to explain matters to colleagues. It’s worth learning how to do that now, when your job/promotion/the future of the universe does not depend on it.
• It is an excellent beginning to the holidays, I beg your pardon, I mean the revision period between Michaelmas and Lent term. (I see that even Part III handbook is now explicit in repudiating any idea that that period might be devoted to carousing with friends and family or watching telly through half closed eyes after too much Christmas pudding.) Preparing a talk is a splendid way of revising. Listening to all the other talks is much better than nothing or even trying to study on your own (who is going to be doing that at the end of term anyway?). I believe this is actually a physiological phenomenon. As with explaining to colleagues, figuring out how to summarise, how to illustrate ideas with examples, choosing good examples, all these activities require using the ideas, and engrave the ideas in your mind in a way that merely reviewing lecture notes cannot achieve.
• It is a good preliminary test for whether you might like doing research in mathematics.  Research does involve some further learning, but going to lectures and doing well in exams is never going to write a thesis for you, however good you may be at exams. Turning the question round, asking what did you get out of a course, rather than parroting all that was in the course is a good exercise developing the skills required for doing creative mathematics. If you are excited by an idea, If choosing something that interests you, playing with it, finding neat examples and explanations that are off piste, working to explain these to your colleagues, that’s a good omen for having fun with research in the future. If playing with ideas in this way does not give you pleasure, think hard why and whether you want an academic career. There are lots of other entertaining things you will be well qualified to do.
• It’s fun.  And there’s a party at the end of it, and if you have given a talk and been to other talks you will feel like you have really earned the party and the food and wine will taste better for it.  

Good, so you’ve signed up to give a talk.

Now, what to talk about. Go for a walk. Think about what ideas were fun. Come up with an example of that idea. Work through the example. An exciting result? How much can you simplify the proof in a simple case? Go for something simple; explain rather than merely recite, have fun with it. Keep it simple, (there, I've said "simple" three times, it must be true) give yourself an easy ride - it’s been a long term.  

Start preparing the talk at least a week before you will give it. Two days before is not enough; you will end up having a miserable experience which does not do your talents justice.

I’m not sure whether someone still talks you through the process of preparing a talk. I used to. Many students think they don’t need advice on preparing a talk, even if they have never tried to prepare a talk before.  They imagine that talking is a natural talent that they were born with. Well, they weren’t born with it. (I’ve just been admiring my 12 week old grand-daughter. Her conversational skills are limited although desire to communicate is very evident). Dan Kan taught me, and for any who will listen, I will pass it on. I’ve left my talk on preparing talks on my homepage for anyone who wants. 

Have fun with it.  It’s part of being a mathematician. 

————————————Story time.  Continuing from the last episode in which the aspiring algebraist made a serious mistake.  Fast forward a year, autumn, the leaves are in full florid autumn colours, there is a nip in the air in the mornings, and a thick litter of abandoned thesis attempts is gently transforming into compost on the floor under my desk...

Usually the greatest benefit (to the speaker) from preparing a talk is a direct result of the effort of preparation. It is surprising what comes clear as the time when one has promised to explain it to someone else approaches.  On occasion however, the talk does generate responses which have profound consequences.

October, 1977, Vermont.  MIT owns a modest country house a couple of hours’ drive north from Boston, and the algebraic geometers were in the habit of renting it for a weekend in the woods. Four faculty, a dozen or so students, a walk in the hills in the morning, talks by us graduate students in the afternoon and evening, tossing a frisbee about betwixt and between.  I can’t remember if the geometers were given a set topic, but I do recall that I listened to a lot of talks about Hilb and Chow, which meant little to me, having no idea at all what these creatures were.

I too had to earn my keep by providing an hour of after dinner entertainment. As I may have mentioned, I did not understand anything of what my supervisor had set me to work on.  I had nonetheless been very diligent struggling with ideas of my own invention. Thesis attempt 33b, great hope of the moment,  was to create a sort of super K-theory using supermanifolds to create a homology theory as vector bundles are used to create K-theory.  This was not entirely implausible; the category of supermanifolds I was working with contained a subcategory which was essentially vector bundles. I had had a go at making some definitions, proving some properties. The big question though was whether one would learn anything new in this way.  I had no idea how to answer it.  However, describing the objects, setting the stage for the question was sufficient material to entertain algebraic geometers for an hour. Should be able to give a harmless talk that will satisfy the requirements.

It had been a very pleasant day.  We had gone for a walk.  We had seen a deer.  The food was excellent.  We played a word game at lunch: how many steps does it take to get from Hilb to Chow replacing one letter each time to get another legitimate English word (I won, although the legality of "holt" was disputed, and defended - eg A.E. Housman, On Wenlock Edge).  Presumably those who knew a bit about Hilb and Chow knew a good deal more by the end of five talks. By the time I gave my talk, the sixth of the day, the audience, replete with wine and dinner, were in a suitably somnolent state for listening to something they didn't really want to know about. It passed off almost without comment.

There was one comment.  Mike Artin scratched his head and said “H^1(Aut), should be easy.” “Ok, I said, you tell me what H^1(Aut) is and I will tell you whether it is easy or not”

Easy for an algebraic geometer, and not beyond me, although the computations were unpleasant. The answer: all supermanifolds in that category were indeed isomorphic to vector bundles.  Sad, I thought to myself, but there you are. Mathematics research is like that; even then I had learned that one’s first guess, however attractive, is not necessarily true (it was attempt 33b at a thesis), on to 33c or even 34.  

However, the story was not quite over. TBC.