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I've been invited to give a 20 minute talk at a mathematics conference in a couple months. The audience members all work in my field, so they will be familiar with most of the terminology and background. I have plenty of material to pull from a preprint I've already submitted to journals, and it seems I have two reasonable options:

Option A. Present everything to communicate the over-arching theme of my research. I could provide several very minimal proof sketches along the way.

Option B. Present one very interesting result (with its lemmas), with more detailed proofs. I could then "sprint" through some related results at the end, or avoid them entirely.

Which will be most valuable to the audience members? Which will make me "look the best"?

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    The other answers have been good, so I'm not going to make a full answer just a comment, but it may help to ask your adviser (if you are a grad student or undergrad) and it may really help to talk to the conference organizers to see what they think would work.
    – JoshuaZ
    Commented Dec 21, 2018 at 23:16
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    I'll add that you need to confirm the speaking time. Usually speakers also have some time for Q&A after the formal presentation. So if you have 20 minutes total, then plan to speak for 15 and allow 5 for Q&A. If no questions then you perhaps can make some additional points for a couple of minutes.
    – MaxW
    Commented Dec 22, 2018 at 3:06
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    Since you seem unsure of this I’m going to assume you’re new at this. I’d say focus on a particular topic unless you’re very senior in the field. These kind of retrospective overview talks are generally reserved to invited plenary speakers. If you attended a conference before, try to emulate the structure of the best talks you’ve seen. It’s pretty common for speakers to choose a topic most relevant to the audience (even if they work on other things too).
    – Tom Kelly ケリー・トム
    Commented Dec 22, 2018 at 3:51
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    I recently gave exactly this kind of talk. I centered my talk around an important tool I use and which is quite nice (nothing too sophisticated). It went great, mainly because of the simplicity of the presentation. I think that in this setting, just as Samuel's spot-on answer suggests, less is more. Presenting one interesting result will draw enough attention for you to discuss later on everything else in more detail with whoever is interested.
    – user347489
    Commented Dec 22, 2018 at 4:57
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    Let me comment that it's possible to be too ambitious with either option A or B. When mathematicians give their first presentations, there is a tendency to try to squeeze way too much material in. Give a practice presentation to your colleague or your advisor to make sure you haven't succumbed to this temptation.
    – Peter Shor
    Commented Dec 23, 2018 at 20:47

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If you have one very interesting result, then I would go with focussing the talk around that. The alternative of trying to cram multiple topics into a 20 minute would be very difficult because that's not a lot of time. Your talk is more likely to be remembered if you present one very interesting result.

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  • As someone who has very little experience with this kind of thing, I think this is what I would most enjoy as well. Something like using one very interesting result to maybe present your method, and then at the end presenting a few things that can also be achieved as consequences of the method or results.
    – DreamConspiracy
    Commented Dec 22, 2018 at 1:51
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I'm going to assume that you aren't an experienced teacher or you might already know the answer.

The tendency is to try to add too much and wind up being overly detailed and pedantic. If you want to give a detailed presentation, then use Option B. Don't depend on having any time to sprint, though. But you should have something to add if you finish early.

However, if you really want to give people insight into what you do, rather than detail, then you could use Option A, but leave out the detail. This one is harder to manage unless you have a lot of insight yourself and can speak in meaningful generalities.

Either of these would be meaningful to your audience. However, if you expect to be facing a number of doctoral students in your field, giving insights is probably more value to them. An exception would be if one or more of the proofs in your presentation is especially interesting in its own right. Interesting proofs are usually more important than the theorems they support as they can open bottlenecks in other research.

Which is easier, depends on you. Which makes you "look best" is impossible to say.

Try to save time for questions. Doing a bit less is probably better than trying to do too much. People can always follow up later if you give them contact information.

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    +1 for discussing the options. I'd add that among all the talks I've attended, both in my particular areas and more generally, I found the overviews that offer insight are the much more valuable and enjoyable. And if your audience misses some small point they are not lost for the rest of the talk.
    – Ethan Bolker
    Commented Dec 22, 2018 at 2:22
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The goal of such presentations should be to interest the audience in your work.

If you fail to do this, your interaction with them will be limited to the 20 minutes allotted.

But if you succeed, they will likely seek you out later, and you will have ample opportunity to expand on the work.

Both of your options are viable: journey highlights and ‘curio’ examination can both be enjoyable. Imagine yourself in the audience, at a time before you did your research. Then pick whichever option would make you want to invite the speaker for a coffee to discuss more.

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An important part of a talk is to introduce your topic and subfield. You need to convince them why what you’ve done is important. Even specialised conferences have broad audiences who may not be familiar with your techniques or application. Even if they are your interpretation of concepts and your assumptions should be stated clearly since people from different places may not follow the same conventions. You will need to explain them in more detail than you would to people familiar with your progress. No matter the conference, you should not assume that they’re already familiar with your topic: it’s your responsibility to explain it (of course can adjust your content to your audience).

You need to get them interested in your topic, tell them why it’s important, why you took the approach you did, and what was challenging or novel about it. A clear structure is essential for any talk. You need to allow time for this.

Considering this, 20 minutes is not a long time slot. You should focus on one or two key results and explain them well. We all do far more work than ends up in our papers or talks and it’s understandable that you want to show your work. However, it’s more important to communicate the key findings well. Especially, if you’re early in your career, you should aim to demonstrate that you know what you’re doing and have mastered particular techniques. If they’re from a related field, they probably have an overview of your area already so a key example to demonstrate why they should care about your area is going to be more interesting for them than yet another overview.

If you race through everything you’ve done, you won’t stand out and know one will understand what you’re talking about or why it’s important. You can of course mention the other research directions that you are working on to put it into context of what you’ve done and what you’re doing next but you need to focus on something. Every PhD student or postdoc has a mountain of data. What your future employers and collaborators are looking for is someone that understands what they’re doing and can communicate it well.

It’s also considerate to finish your talk on time. Allow plenty of time for questions and discussion afterwards. If you go overtime, you will have to rush and this will not help your nerves. You’ll be very unpopular if you cut into the time of the person after you and delay everyone’s coffee break. Keeping a conference on time is difficult and adhering to the schedule is important. It’s a lot easier to keep to time if you set achievable goals of a few key points. You need to identify the most important findings to present or emphasise: this is a skill you should develop in a career as an academic researcher.

Help yourself and your audience by preparing well. There’s only so much that people can remember for take home messages from a full day of talks anyway. Think about what you want them most to understand from it. If they want to know more, they can discuss it with you after your talk or read your papers. The purpose of a talk is really to introduce yourself and give people context and background.

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  • You still need to put your work in context, especially if people may join from another parallel session (as they often do). Jumping right in to the details is exactly what I take issue with in departmental talks. You cannot assume that that they’re familiar with the topic already: there are many different subtopics. If you cannot explain it well to members of your own department familiar with your progress, you need to rethink how to introduce it to people from beyond who’ve never seen your work before.
    – Tom Kelly ケリー・トム
    Commented Dec 23, 2018 at 15:31
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    "Even specialized conferences have broad audiences who may not be familiar with your techniques or application. You will need to explain them in more detail than you would in your department." Where does this come from? If everybody giving a talk in a special session on, say, Geometric Analysis and Geometric Flows gave an introduction that was broader than what would be needed in your math department, half of each talk would be repeating information that every other single talk in the special session contained. A big waste of time for a very short 20-minute talk.
    – Peter Shor
    Commented Dec 23, 2018 at 15:33
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    In math departments, you can't assume that other members of your department know anything about your research. The people in his audience are going to have a huge head start over most other people in his department because they already know the field.
    – Peter Shor
    Commented Dec 23, 2018 at 15:39
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    If you're talking at a specialized math conference, or a special session, you should pitch your talk so that researchers in your area, and closely related areas, will be able to understand it well. It shouldn't be so specialized that the only people who can understand are in your narrow area, but it doesn't need to be so broad that it bores all the people in your field. If your audience is much broader (say a colloquium at another math department), you shojld pitch your talk so most mathematicians can understand it, but it will probably be a lot longer than 20 minutes.
    – Peter Shor
    Commented Dec 24, 2018 at 21:43
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    Your answer is much better now.
    – Peter Shor
    Commented Dec 25, 2018 at 0:47
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Communicate the theme (A) sans the "several". Pick one or two of most interest. 20 minutes is not much time.

Remember people are deluged with different micro papers and talks. If you give them a general feel for an area that is more benefit (even for experts).

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