First, addressing slides or boardwork:

• Don’t put too much on a slide.
• However, don’t put so little on each slide that you’ve moved on to the next before the audience has time to read it.
• Feel free to only say some things. In fact, if literally everything is written out the audience may wonder why they’re listening to you when it would be much faster to read the slides. You need the key points, the technical definitions, and anything else the audience may want to re-read, as well as diagrams or pictures. Think of your slides as the highlighted or boxed portions of a textbook.
• It is helpful to repeat important information (especially definitions) on later slides, or at least verbally.
• Turn down the contrast and brightness on your computer screen (both separately and together) to check readability if you’re using anything other than black and white; projectors often wash things out a bit.
• In a chalkboard talk, write top to bottom, left to right, and respect the seams of the board as edges of paper – it is difficult to write across them neatly. When going back for a second round, erase your previous writing thoroughly.
• Make sure the type (or your handwriting) is large enough.
• Use whitespace generously – avoid large blocks of solid text as much as possible.
• Landscape orientation allows higher magnification since it matches the usual projection screen dimensions better than portrait.
• Stand to the side of your slides or what you’ve just written. It is continually surprising to me the number of people who stand in front of their writing until they have moved far enough past it in the talk that reading it would be a distraction to the audience instead of a help.
• Proofread! You might not catch everything, but you’ll catch the most egregious errors. Putting the slides aside for a day will help your observational abilities – when you look at something repeatedly you often start seeing your memory of it rather than the actuality.

Secondly, your speaking:

• Practice! When you have given many talks you will need less practice, but when you are starting out, multiple sessions are desirable.
• I have never attended a talk that was read aloud from a prepared paper that did not bore me to tears. Use notes, but don’t read word for word.
• Speak loudly but do not shout. While whispering and mumbling are clearly not desirable, neither is yelling. Note that you can yell without raising your voice, as well. The audience will not understand more if you are strenuously emphatic; they will simply feel accosted.
• Note that while practicing your talk by yourself is good, it is inexact for timing. Err on the side of “too short” (within reason, of course – prepare more than thirty minutes of material for an hour-long talk); between extra things you say and questions from the audience the length will most likely increase, and if not, well, no one minds ending early.

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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.”

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