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I am a graduate student in mathematics. I am preparing to give a 20-minute short talk for a conference. I am looking at some beamer presentations this one and this one. Neither of these slides neither have any references, nor any numbering of definitions, theorems, propositions, lemmas etc.

I have some few questions in mind:

  • Is it normal to omit proof of the main results?
  • Is it normal to omit numberings of theorems, lemmas, propositions, etc.?
  • Why don't people use \begin{theorem}... \end{theorem} for theorems and the same for lemmas, and propositions?

Besides that, what would be the best way to present a short 20-minute conference talk?

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    Your talk is only 20 minutes long. Putting proofs into a beamer talk most likely results in losing the audience, especially, if proofs use more than one slide and if the audience has to remember formulae from previous slides. Similar for numberings of theorems. May 23 at 14:31
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    And just to complete the picture, I'll add to all the good advice that you've already gotten: Keep in mind that at some conferences there's even the option to use blackboards instead of slides. May 23 at 20:37
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    I am thankful to all for giving me valuable suggestions. All the answers are just equally excellent.
    – MAS
    May 24 at 2:53
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    The Notices of the American Mathematical Society has a column dedicated to advice for early-career scholars—I recommend checking it out for articles like this one by Bryna Kra that directly addresses your question. Indeed, you might ask your advisor to sponsor you for a student membership to the AMS and other professional societies. May 24 at 7:24
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    As someone who works in a math department: in this field, you can get away with anything. Want yo just present your paper in beamer format, no changes to content? People do it. Want to have an extremely didactic (skipping proofs) presentation? people do it. There is no norm. My advice (from a non mathematician like me) is to try to get to the most people in the audience. May 24 at 10:24

7 Answers 7

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Having given loads of presentations, scientific and non-scientific, here a a few nuggets for you to consider

  • figure out who you want the talk to be for: to the famous mathematician in the room you wish to impress, to the professor a university you want your next post-doc to be, to young researchers, ... That will set the tone of your presentation
  • even if for a mathematics talk, how technical it may be, you are telling a story. Find out what that story is and how you want to bring it over to the audience and make sure you do that in your talk
  • what you say is way more important than what's on your slides. Your slides should be a invitation for the audience to listen to you. So keep them as simple as possible. If they want the nitty gritty they can go to your paper(s)
  • at the start of your talk, spend some time talking about you, your work, the weather, or whatever. This may last only 10-15 seconds and is just to make sure that people get accustomed to your voice and your way of speaking. If you jump into technicalities straight away, your risk losing part of the audience straightaway.
  • do not overload your slides; do not use full sentences; do not give definitions of things people know. Avoid at all cost that the audience has to focus more on what's on your slide than on what you say. The worst example of this if you put so much information on a slide that people don't have time to read it. A good way is to write your slides and then delete 50% of what you have written!
  • use colors to emphasise the most important message of each slide and don't put more than one such message per slide.
  • make sure you can adapt your talk to speed up or slow down, depending on how it goes. Practice your talk and if you are the type that overruns, makes sure where you can cut some parts. If you are the type that dashes through, make sure where you can expand a bit. Take off your watch and keep it visible. Have points in your mind where you want to be after 5', 10', 15' and adapt your speed. Avoid at all cost that your are being told you have 2' left and you are still at the introduction!
  • for a 20' talk you need max 10-12 slides.
  • leave a bit of time for q&a and answers questions directly, don't wait until the end. If you don't know the answer think in advance of some standard things you can say. And if you don't know the answer there is no harm in saying this or that this is something you are still working on. If you get a stupid question, you can always say "that is a very interesting question" and just proceed with your talk.
  • if you develop an complex argument during your talk make sure you give pointers to the audience where you are and where you want to go.
  • don't end abruptly; remind your audience of the key messages you want them to take away.

good luck.

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    Great advice. I can't emphasize enough the point that the speaker is telling a story. Whatever it takes to get the audience to understand the story (including lying about irrelevant mathematical points!) is what should be in the slides. We want the audience to leave the room able to tell their friend who missed the talk what the speaker's story was. May 24 at 7:22
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    I have a question at your 9th advice. You said leave a bit of time for q&a and answers questions directly, don't wait until the end. Do you mean to do it at the end of my talk or at the middle of my talk as well ?
    – MAS
    May 24 at 15:50
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    @MAS i Find it better to address a question immediately, but don’t it distract your talk. So give a fairly short answer. If the answer is rather long, you can always suggest to meet during a break. If you get too many questions, then you can relegate them to then end of the talk. May 24 at 17:59
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    @GregMartin I would strongly advice not to lie about anything during your talk. It can discredit all the rest you say. May 24 at 18:01
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    Oh, one other thing. If you can anticipate questions (but don't want to spend time on them unless they are asked), prepare slides for them, and put them after a blank ending slide. That way, you can say "Oh, I thought someone might ask that" and pop up the optional slide
    – Flydog57
    May 24 at 18:27
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It's important to remember that a talk is not a paper.

Is it normal for people to avoid proofs of their main results?

Yes, though it depends on subfield, and it is especially common in a 20 minute talk. Your audience does not have time or inclination to process many technical details in this format.

Is it normal to avoid numbering?

Yes. Your slides are not a paper. Numbering schemes aren't very useful. Do not distract your audience with irrelevant text, use \begin{theorem*} and family.

How should I present my talk?

This is too broad to answer really. But I highly recommend that you (1) make sure to look up how remove the Beamer buttons and (2) Do not include an outline of your talk with the number of slides like one of your examples does. It's distracting and rarely useful. Do have slide numbers on the slides.

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My favorite advice comes from Cosma Shalizi
(his background is in statistics and physics, but the advice applies equally well to mathematics talks):

...do not overload your audience, and do not even try to convey all the intricacies of a complex academic argument in your talk. The proper goal of an academic talk is to convey a reasonably persuasive sketch of your argument, so that your audience are better informed about the subject, get why they should care, and are usefully oriented to what you wrote if and when they decide to read your paper. In many ways a talk is really an extended oral abstract for your paper. (This is more effective if those who are interested can read your paper, at an open pre-print archive or at least on your website.)

So yes, avoid proofs and numberings to minimize overload on the audience. Instead, make it easy for the audience to find your full paper, if you have written it already---or at least to your website where you'll eventually post the paper.

Shalizi's post has further advice on what to present and how to prepare. But if you start with "your talk is an oral abstract for your paper," the remaining details are just corollaries :)

Also, especially since you're not a keynote speaker and the audience isn't there specifically to see you, I also like the advice from Kathy Sierra:

My path to coping with heart-stopping stage-fright is to focus NOT on what I [the presenter] do but on what they [the audience] experience.

and

Open with a question they would very much like an answer for.

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Regarding the specific questions: A 20 minute talk needs to be incredibly focused; there is not much time. Normally, explaining your results and they context they fit into will take all your time, and presenting the proofs is not a good idea. Especially in the 20 minute time frame, you should think of a talk as an advertisement for your paper, not a replacement. (Corollary: Your paper should be on the arXiv before you speak!) That said, if the key new idea that you are bringing is a proof, and if the proof has a key idea, it can work to sketch the key point of the proof.

Numbering Theorems and Lemmas is not likely to be useful. If you are going to refer to a few key results throughout the talk, I would suggest giving them descriptive names instead: "the lower bound", "halfway to Smith's conjecture", "the key regularity hypothesis", etc.

More broadly, I am sure that every mathematician has their own method, but here is how I write a talk:

  • Wander around explaining the result to myself, on long walks or while cleaning the house. Imagine how I might present it to different mathematicians I know.

  • Outline the talk in pencil on a notepad.

  • Make a rough draft of slides in LaTeX, omitting figures, difficult equations, references and complex typesetting.

  • Print out the slides on paper, turn on my stopwatch and give the talk 2-4 times. It is important to actually speak out loud and include places where I will pause and make transitions. Note on the slides in pen any points that I want to add or revise. This is usually the stage at which a lot of material gets cut for being too wordy or complex.

  • Make the edits I noted.

  • Run the talk again to make sure that the edits work.

  • Add figures, equations and references. Tweak slide layout. Make a cover slide.

  • Ideally, return to the rehearsal stage and run the talk another 2-4 times.

I'll often memorize an opening sentence or two, so that I sound confident and say what I want to at the start.

It's been a while since I gave a 20 minute talk; I think the last time was at an AMS meeting in 2017. Here are the slides for the two talks (1 2) I gave there. The first presents a proof; the second (more typical for me) does not.

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    Thank you for your nice answer. I really appreciate that you have attached two of your beautiful slides. It will help me
    – MAS
    May 24 at 2:42
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Is it normal to avoid numberings of theo, lem, prop etc ?

Many already answered that it's normal, but I'd like to add something on why. Numbering theorems and equations is useful only when you want to refer to them later. In talks, people cannot go back and forth freely to check what (3) or Lemma 1 are, if you refer to them a few slides later. So if you write something like "Thus using (3) we prove Lemma 1", and they are not defined on the same slide, you have already lost most of your audience.

For the same reason you won't see numbered references: people cannot skip to the bibliography during your talk to see what [2] is. If they are really necessary, sometimes references are written more explicitly [Einstein '04], or put in footnotes; sometimes there is a final slide with a list of references, or sometimes just a pointer to the Arxiv preprint on which the slides are based.

OTOH, numbering pages is useful: sometimes people in the audience note down the page number when they have a question, so that at the end of the talk they can ask "please go back to slide 13; why is the function f there continuous"?

Why don't people use \begin{theorem}...\end{theorem} for theorem and same for lem, prop ?

Note that \begin{theorem}...\end{theorem} is typeset differently in Beamer: it looks like a \begin{block}{Theorem}...\end{block} with a different color depending on the theme. So it is very well possible that even the slides you have linked use that environment.

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Your first example is a talk about Belyi maps, and it uses the function f(x)=3x^2-2x^3 as a running example. If I were giving a 20 minute talk on that work, I would structure it as a talk about the properties of that particular map, with plenty of pictures. I would not give any precise definitions or state any precise results unless they are very short and self-contained. Instead, I would say things like this:

If we used a different polynomial f(x) then we would get a different pattern of permutations, and various other aspects of the story depend in a well understood way on the cycle types of those permutations. There is a good general theory about this which is explained in the preprint.

In general, explaining a single well-chosen example is an underrated approach for very short talks.

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A 20 minute talk is best thought of as an ad for your paper. The goal is simply to state the main result and for the audience to understand what the statement means and why it matters. This requires giving some definitions, some context, and maybe an example. It’s ok to just end the talk after giving the statement, or if you want you can have one more slide talking about further directions or saying a sentence or two about what kind of ideas go into the proof. You will not be able to go into the proof beyond one or two sentences in a 20 minute talk, and it’s totally fine to spend zero sentences talking about the proof.

If people want to see the proof they can read the paper.

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