I started compiling advice for conference and colloquium talks some time ago, writing notes to myself while attending talks, prompted by good and bad features of the talks I was listening to, as well as from my own experiences. This installation of advice is aimed at the content of the talk; next week I’ll have one aimed at delivery. Note that since I have primarily lived in the technical world, this is aimed at people giving technical talks. The next installation will have more points that are applicable to talks in general.
The key to giving a good talk is to remember that talks are about the audience, not you. If the audience comes away understanding your result and its significance, you win. If the audience comes away confused, annoyed, or otherwise feeling that you’ve wasted their time, you lose.
I always tell students that talks are an exercise in letting go. You must accept that you won’t be able to say everything you know. There’s a time limit and a speed limit, and both need to be respected to keep the audience’s respect. Pick a goal for your talk (understanding the statement, motivation, and significance of one specific result would be a typical one) and ruthlessly tear out anything that does not assist you in getting to that goal. Omit proofs if you can and give high-level summaries or “proof by example” if you cannot; give only special cases of definitions and results if that’s all you need (but always label them as such).1 Less is more.
Your audience will be more forgiving of seeing several things they already know than of an unfamiliar idea being flashed before them with inadequate preparation. Take into account the audience you are going to address, of course: a seminar in which everyone is at least a graduate student in your field requires less background than a colloquium talk wherein, although everyone is a mathematician, many will have seen your topic most recently in a class they took a decade or more past. Within reason, it’s all right to be imprecise, as well. I have seen many talks where the speaker said something along the lines of, “I’m lying to you now, but it’s morally correct.”2 Examples are helpful and getting bogged down in details is not; it will require more than a talk worth of information to truly understand the details of your work no matter what you do.
Give context for your work. As with background, how much you have to say will depend on your audience. The history of your work can be helpful in understanding how you came up with your proof, if it is not a standard technique (e.g., “we tried this and it didn’t work, so we looked into why…”). Clarify your contribution but don’t obsess over it; while you certainly don’t want to appear to take credit for someone else’s major idea, you don’t have to itemize who worked out each specific detail.
Think carefully about organization. The best order to explain things in a talk may be different from the best order for the full research paper. Remind your audience of your goal periodically and repeat key definitions or lemmas if they are being used any time other than immediately after their original statement. Don’t insult the intelligence of your audience, of course, but remember that most or all of them haven’t been working on this topic recently, unlike you, and can’t refer back to earlier material like they could in a paper.
The 3 Cs are to be clear, concise, and charming. I have no advice for the last one, but I hope this post has been useful for the first two.
1 Of course in a topic-specific seminar, you may have been asked to explain a proof, in which case this rule is clearly changed.
2 Any mathematician will understand this phrase, but non-mathematicians might not. “Lying” is used typically to mean “being somewhat imprecise,” and “morally correct” means “literally false, at least in some aspects, but gives good intuition.”