Advice for talks

Goal of taking part in a seminar

One could argue that the best way to learn a concept is to explain it to someone else. Nonetheless, a seminar has other goals, such as practicing the following skills.

• Writing skills: while writing a report, you learn how to use the appropriate mathematical typesetting tools. While doing so, you are also exposed to the standard conventions for written maths.
• Presentation skills: giving a presentation requires taking care of many problems at once and dealing with how they all interact with each other.

In this document, I am not going to touch upon the first point, so here are some pointers in case you are interested in it.

• Keith Conrad’s advice on mathematical writing.
• Olivier Brinon’s décalogue à l’usage de la rédaction des mémoires.

In the rest, I will deal with what contributes to a presentation. I will concentrate most on the preparation of the talk. For the presenation itself, I will limit myself to some small advice and pointers. This topic would deserve its own in-depth document, this is not it.

Preparing the talk

Let’s go through, essentially in chronological order, what steps one needs to tackle to prepare a talk and what questions one should keep in mind while doing so.

Reading the material

While reading the material, keep the following questions in mind.

• What is the main goal of this talk?
• How does this talk relate to the rest of the seminar? (For a talk outside of a seminar: why is the talk interesting to the audience; how does it relate to their interests?)
• How do I tell the story of my talk? More explicitly: what needs to go into the talk and how do I organize the material?
• What is the set of prerequisites for this talk: what can I take for granted, what should I spell out in full and what should I recall from previous talks?
• What content is more important and has to be included at all cost, and viceversa what is less important and can be skipped if time is running out?

When reading about a new concept or definition, I suggest the following questions.

• Why was this definition introduced? What purpose does it serve?
• How can I intuitively understand this object?
• What are some interesting examples for this object?
• (Especially valid in algebraic geometry): how (and why?) has the definition evolved over time? Is there a more classical definition that can motivate it?

I advise these related questions for results and statements.

• Why was this hypothesis introduced? Where is it used? What happens if we drop it?
• Is this result surprising or expected?

Redacting the notes/report

You should start writing once you have a clear(ish) idea of the story of your talk, but before you feel that you understand everything.

As a general piece of advice, you should write as if you were addressing yourself, before you started learning the new material. For talks, you want to adapt the level to your audience. For students seminars you can use the set of prerequisites (and the rest of the seminar content) as an accurate indicator of the level to aim at.

Especially if you type digitally, please mind the conventions of mathematical writing linked above.

Moreover, try to keep an eye on the following.

• If there is more than one source, try to uniformize notation in your notes. Ideally, you would choose what is currently standard in the field.
• If you decide to change any definition, try to go through all of the results where it appears and check for any side-effect of such change.
• If you invoke a piece of notation or an object, make sure that it has already been introduced (either in your report, before you use it, or in a previous talk).

According to my experience, the best way to make sure to comply with the above is to let the script sit for a couple of days (best would be weeks) and to go back to it with a fresh mind (which means “having forgotten most of the details”).

The preliminary meeting

This section only concerns students seminars.

Once you finished typing your notes, you should send the report and book a preliminary meeting with the organizer(s). In the meeting you want to ask about any detail that still confuses you, be it mathematical, historical, motivational or of any other kind. Moreover, if you could not find an answer to any of the previous questions, this is the perfect occasion to ask them!

Finally, if you managed to send the report to the organizers in advance, you will get some feedback about the mathematical content and style of the writing in your report.

As a rule of thumb, I don’t expect students to be able to answer all of the questions listed above. It is therefore almost guaranteed that you will bring something out of the preliminary meeting. From experience, it tends to be an extremely productive moment.

The presentation

Notes

I suggest to go to the presentation with some notes. You can either use your report, or some other notes. In any case, I would suggest to write down all of the content that goes on the board and anything else you might be afraid of forgetting. (This could mean historical notes, names, instructions to yourself, etc.) You don’t want to find yourself in the situation where you don’t know where something is or how to write down a definition.

You might also want to write down spoken parts, especially if you find it tricky to recite them correctly and they cover an important role in the talk. (An example could be the introduction to a topic you are not too familiar with, in your second language, while in front of your supervisor/famous expert in the field.)

You don’t want to just read your notes. I would suggest to keep them as a guide, to remind you of the order of the topics, and to use them when writing definitions or more complex arguments on the board.

Pace and writing

Concerning pace, you want to give some time to your audience to digest new definitions and manipulations. It is not always easy to figure out whether the pace is adequate, but I suggest to prefer a more leisurely pace when possible.

As a rule of thumb, it is better to write down more on the board than less. It slows the pace down, and it allows to access information at a later point. At the same time, sometimes it is convenient to only say things out loud. Here is a small list of examples that could help approach this problem.

• Definitions and statements: always write down in full.
• Hard algebraic manipulations: always write down in full.
• Sketched arguments: write down just key points.
• Historical notes: either write down key points or nothing at all.

Keep in mind that, the less you write, the more importance every symbol you write gains. Write tersely only if you are confident that what you are writing is readable, clear and consistent. In such case, be very deliberate with your word and symbol choices.

Try to pay active attention to this topic, since there is no universal answer, and you might want to adapt your strategy to the kind of talk you are giving.

After some practice, you might find your notes to be a very accurate prediction of how much time you will need for the desired material.

Various advice

Some extra, unordered suggestions for the talk.

• Start the talk by giving an overview of what is going to happen and by explaining how the talk fits in the context of the seminar.
• Some advice on how to use the time it takes to erase the board: you can give an overview of what is about to come, motivation for something that just happened or is going to happen, you can ask for questions or even just give time to the audience to process the previous content. Any of them is a valid choice, in the correct setting.
• Differentiate lower-case from upper case letters also in speaking. So, \(x \in X\) would be spoken as “small x in (capital) x”. (In such situation, please avoid saying “x has the following property”, as it is not clear if you are referring to the element or the set.)
• When introducing a new concept or result, try to present as many answers to the questions you already asked yourself.
• Clearly state when you are done with the talk, as sometimes it could not be obvious from context.
• Keep track of time to adapt what you are going to present. Aim to be done before the time is up, not after.
• If you just started out giving talks or need to give a particularly important one, you should at least practice it in advance. If possible, at a blackboard (or with a beamer). Even better would be to record it. The best is to watch it back and practice the advice on attending other talks on yourself.

Attending other talks

This piece of advice applies more in general to attending talks in any seminar, symposium, conference, etc. Of course, you probably want to concentrate most on the content, but I would suggest to try to take notice of other aspects of the talk too.

Interestingly, it is possible to use other people’s talks as a great learning tool on how to give a talk. Independently on whether you enjoyed it or not, you can always ask yourself what made that talk great or not. In particular, when you are not enjoying the presentation, ask yourself what you would change to improve it.

On this note, you can start practicing this skill by reading the following (hilarious) article of Evgeny Shinder on how to give a bad talk or the similarly themed talk of Jean-Pierre Serre on how to write mathematics badly.

To expand upon this section, let me link the great advice from Ravi Vakil for potential PhD students. Everything on that page is worth reading, but the sections “on seminars” and “on giving talks” are particularly relevant to the current discussion.

Pointers

Fathi Ben Aribi

50 tips for preparing and giving math talks. Here you can find great advice on how to approach preparing and giving a talk, especially suited for research level talks, but you can still find some great suggestions.

Agelos Georgakopoulos

How to give a talk that is not too bad. Here you can find some great extra advice on how to behave during the talk. There is also advice for preparing the talk, but it is aimed at research-level talks.