Of course you're going to give a Part III Seminar talk

It's about the Part III Seminars again.  If you are doing an essay (and of course you are), and if your essay topic did not depend critically on topics covered in the last four weeks of a Lent term course, then of course you are going to give a talk.  Never mind the fact that it is a fantastically inconvenient time, that you haven't really understood the paper yet, (but would have time in the Easter hols study period to get a lot further with the subject),  that you haven't had a full night's sleep since sometime during the early part of January, and that you are nearly four weeks behind in copying out your lecture notes for this term.

Actually, we did think very hard about the timing of this seminar series, and every time we thought about it, we came back to the conclusion that it was the right time, inconvenient as it might be. Here are a few of the thoughts behind our inconvenient choice.

• Giving a talk is a really important "first draft" for any paper you might ever write. Particularly if you obey the command to write nothing, but start from a focus (see notes on how to prepare a talk*) and just begin at a blackboard in an empty room and start talking. This is how you find out whether you understand, what you understand and what you don't.  It is almost impossible to stand up and spout codswallop out loud in front of an empty room.  The walls snigger at you, motivating you to go back to an example maybe, and try, really try, to figure out what is going on. It is also the most efficient way of ensuring that your notation is consistent and your logic not circular.
• If at all possible, getting this "first draft" done before the start of the Easter study period is a really good strategy.  You can then spend the first week of Easter study period writing the essay as if it were merely a long take-home exam, splattering the words on the page* as fast as possible, and then for the rest of the study period alternating revising courses and revising the essay - two very different types of work so almost, in a way, relaxing.
• But, you may say, I really don't understand it yet, and I need to get to the main theorem before I can give a talk. To which, I would reply, do you really? If the main theorem is still some kilometres distant, there is probably some interesting terrain that you have covered, and here we are kindly providing you with an incentive (aka gun at head) to get to grips with that terrain.  Find something about that mathematical landscape to explain; that at least is an introductory section to the essay Dealt with. Moreover, if things go pear-shaped, if you don't get to that promised result, in discussion with your essay setter you may discover that a really good presentation of that terrain might earn you more marks than a badly present main theorem.

This last issue, the decision whether to cut your losses and give a talk on what you know now, or press on to understand a wee bit more will pursue you throughout your academic life.  There will always be something else to understand.  However, mathematics needs to be done and it needs to be communicated or it may as well not have been done.  Your effectiveness as a mathematician will reflect some non-linear combination of your ability to come up with interesting ideas and your ability to share them.  Coming up with the interesting ideas is often a matter of luck and inspiration, and as such cannot be conjured or commanded. Anyone can learn to present well, and learning to call a halt to the process of pushing investigations further is an essential first skill to master.  Master it.  Give the talk. 

Let me strengthen that last point.

• Pursuing further investigations rather than writing up now what they know already is a recognised way by which good students in particular get disappointing Part III results. By this time in the Part III year, a creative student, creative students in particular, will be heartily sick of sitting in lectures. (Cheer up - not long now and you will never do it again!)  Spending time thinking creatively about an essay becomes increasingly attractive, particularly in contrast to revising courses containing a fair amount of material that you don't really want to know. Marks in exams, particular on the less than favourite courses may suffer. Moreover, the stop investigating start writing deadline gets pushed back and pushed back.  The incentive to prepare the spoken "first draft" is lost, time becomes short, the spoken "first draft" gets omitted. Without the walls sniggering, the task of editing out the codswallop requires unnatural self discipline.  Even the essay mark may prove disappointing.

A short story to strengthen the case for taking the time to think carefully and present the "terrain" even if the goal remains elusive. A student of Dan Kan's, some years senior to me at MIT was growing old in his status of PhD student, and progress towards a thesis had stalled. Somewhat unsympathetically Dan made it clear that the young man's best option was to write up what he did understand, submit it a a master's thesis and get out. When it comes to that point, a student has few options, and the young man did as he was told. And came in one memorable morning in a highly excited state drunk with understanding - the methods he was attempting to describe in fact generalised nicely.  QED PhD.

Understand something.  Give the talk.

*These notes were written ages ago.  I am not likely to get around to revising them, but my opinions haven't changed, and revisions in any case would be mostly cosmetic.