Ravi suggested that I should give a stable bloghome to this short .pdf of tips for giving math talks, which I wrote a few years ago for our graduate student conference in number theory. It’s aimed at people giving their very first seminar talks. Readers, please add in comments the advice I forgot to put on the tip sheet!
Update: A reader helpfully points out that I basically already wrote this post, less than a year ago, and linked to the same tip sheet. Sorry! I have a little baby! I’m sleepy and I forget things! Anyway, the link in the old post was dead so at least the repost serves the purpose of making the tipsheet stably available.
Quomodocumque 
Some very basic pieces of advice:
Wear a watch.
If you are going to use chalk, wear a white shirt.
A few more:
If you plan to write anything complicated on a blackboard — a large equation, a commutative diagram, a drawing — draw it ahead of time on a piece of paper so you can copy it exactly. If it is very complex, consider whether you could draw it before your talk starts, or bring it in on an overhead.
JSE’s advice to practice your talk in front of another person is very good. A good second best solution is to practice your talk in a room by yourself, speaking all the words out loud, and writing and erasing from a blackboard everything you will write and erase during your talk. I usually find that a 60 minute talk compresses down to 40-45 minutes when I rehearse it this way; if it takes you 60 minutes with no audience, then it is definitely too long.
Decide ahead of time what you will cut if you start running behind on time. Maybe it is the last 5 slides, but more likely it is a proof or technical lemma in the middle of the talk. Definitely do not go hyperspeed through the last 5 slides in the last 30 seconds (or, as you said, go over time).
In the book “Made to Stick” they talk about the “curse of knowledge.” You are the expert and you REALLY want to tell your audience everything you know—not only the theorem, but the cool corrolaries, the neat example you discovered, the counterexample you found to your earlier conjecture, the history of the problem, the false proof that so-and-so gave, etc. etc. You should definitely include some of these things, but if you include everything you’ll go over time and lose the audience’s interest along the way. Force yourself to cut things. (Paraphrasing that old saying: “Sorry I gave a long talk, I didn’t have the time to prepare a short one.”) You may be able to bring out some of these tidbits during the Q&A period if the audience wants to know more.
My experience is that many people giving their first talk simply aren’t well-enough prepared. It’s not obvious, until you do it yourself, just how many hours of work can go into preparing a good one-hour talk.
After attending a bunch of rather shambolic talks by graduate students, I wrote my own short list of tips. It’s not as good as Jordan’s, but puts more emphasis on what the constitutionally disorganized need to do.
It might seem overly prescriptive. On the other hand, when you’re doing something for the first time, rules can be a safety net.
This is an excellent set of tips. I think though that a short bibliography placing the talk in some historical context or providing further reading for definitions or results glossed over in the talk would sometimes be nice. The last slide of a talk that I saw recently was just such a short bibliography, and I appreciated it. The blackboard is not a good place for this, but in that case the speaker could bring a few single-sheet printed bibliographies for those who might want to take one while walking out. Handouts would even be a good idea for those using slides. You might want to consider though whether or not you could feel crushed if no one wants your handout.
As well as putting effort into the formal aspects of preparing a talk (timing, structure, etc.), another place to put effort is into thinking about the mathematics. Indeed, I have seldom put effort into the former (and while I have given some bad talks, I think I’ve given some okay ones
too), but I frequently put a lot of effort into the latter.
I don’t mean here the work of proving the theorem and so on; obviously we are all putting effort into that (!), and that work has been done before you are giving the talk. (Presumably, it is the
basis for the invitation to speak.) What I mean is thinking about how to explain the mathematics;
what metaphors to use, what order of logical development to follow, and so on. I think that there are lots of ways to explain ideas, even very standard ones, and that, generally speaking, there is a lot of scope for improving mathematical exposition by being more imaginative in this regard.
I certianly don’t have anything grandiose in mind. To confirm this, let me give just a couple of examples. (I will just draw from my own field
(since this is all I’m really qualified to do.)
Ex. 1: When discussing Spec A (for some ring A) with non-experts, rather than mentioning prime ideals or other things that are a complete mystery to non-algebraists, one can emphasize that A is a ring of commuting operators (never mind on what), that Spec A is their simultaneous spectrum, that if M is an A-module then the support of A consists of those points in the spectrum which have a non-zero eigenspace in M (the latter is a lie; one should co-eigenspace — but this hardly matters for a non-expert), and so on. I think that spectral theory is more familiar to mos mathematicians than commutative algebra.
Ex. 2: When discussing deformation spaces for Galois representations, one can described them as representation varieties for the Galois group, rather than trotting out functors on Artinian rings,
universal deformation rings, and so on. Representation varieties of groups of various kind are
more widely known (I think) then functors on Artinian rings.
These are certainly pedestrian, but the reason I mention them is that they are technical aspects of my own field which, in my experience: (a) are *extremely* unfamiliar to most people out side the field, and which I’ve seen cause eyes to glaze over in audiences of talks in which they appear, and which (b) admit explanations which are less technical than one usually sees, but are also quite honest as explanations (more so than they might appear at first), and which bring out the connections with other parts of mathematics.
So maybe I’m thinking of something more specific, in the end: it would be good if we all thought as much as we could of how to explain technical aspects of our work in terms that are familiar to as wide a range of mathematicians as possible (even if these terms are not quite the standard ones we use in our own area).
Great idea for a blog post. Two comments…
1. I liked how Jordan’s advice and Tom’s advice are different. In fact, there were bits of both that I wasn’t sure I agreed with. But I think there are very few rules that everyone will agree on (e.g. going over time), so it’s much better to take the union and have lots of contradictory opinions than to take the intersection and have a small number that everyone can agree on.
2. Following Matt’s comment, I think it can’t be emphasized enough that the most important thing in most talks is to actually explain something to the actual people in the room. To do this, you really need to think about two things: what you want to explain, and what mathematics the generic audience member is comfortable with. In my experience, putting yourself in that frame of mind is really helpful, perhaps even imagining yourself explaining it to a particular person. There are other situations (job talks, plenary talks,…) when there might be other important things to do, but even there, people will be really happy if they actually learn something they can take home.
Nice list Jordan. I’ll have to point students here.
“two things: what you want to explain, and what mathematics the generic audience member is comfortable with”
I think it’s important to keep in mind that there’s no generic audience memnber, though! The talk has to be constructed so that the graduate students in the room will get something out of it, as will the experts, as will the mathematicians outside your field — but not the SAME thing.
Good point! I do think there’s some risk in spreading yourself to thin if you try to give a lot to everyone, but if you can pull it off, so much the better.