I am a graduate student in math (algebraic geometry) and a course I did was simply self studying a 600 pages textbook on algebraic geometry and discussing it with my advisor. As part of the assessment for this course, I need to give a 20 minutes talk. The audience are other graduate students and academics; the audience may not be algebraic geometers. My talk gets marked by academics and not students.

My problem is that I have no idea how I should give a talk about a 600 pages textbook full of technical details to a general math audience. What should I include or exclude? What is the usual practice? How do people usually give such talks? Please note that my audience may not be all algebraic geometers. I asked my supervisor and the only answer I got was, "Just give an interesting talk!"

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    What was the exact wording of the instructions or assignment? Jan 5, 2022 at 22:24
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    When you say the audience includes graduate students and "other academics", are the academics all going to be mathematicians at least, or could you could have a linguist or psychologist in the audience? Clear this up in your post.
    – KCd
    Jan 6, 2022 at 0:02
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    What was the book? Jan 6, 2022 at 20:55

9 Answers 9


First of all, let's be clear about what the purpose of this part of the course is. It is not to demonstrate that you have understood the material in the book – I imagine that your advisor already knows how well you understand the subject matter if you have been discussing it with them throughout the semester. People will not be sitting there trying to catch you out on some technicality. It is to give you an opportunity to learn how to communicate math to other mathematicians, which is a crucial skill for any mathematician to learn and it cannot be acquired without trial and, inevitably, error. The most fruitful way of taking advantage of this assignment would be to put any worries about how the talk will be graded to the side and think of this as a learning opportunity.

Secondly, let me suggest that it is not terribly helpful to think too hard about what you should include or exclude or about what the usual practice is. Instead, you can mentally frame the task in this way: you are being afforded a very scarce resource, namely people's largely undivided attention for the duration of 20 minutes. In return, people would like to learn something of interest. (Although in this case they will hopefully understand that this is more about you learning how to communicate mathematics than about them.) Therefore, I propose that you focus on the pay-off for the audience instead of real or imagined ideas about what a talk should or should not include. This, perhaps, may be what your supervisor meant when they told you to "just give an interesting talk". In a nutshell, you should ask yourself: what will a mathematician who does not know a whole lot about algebraic geometry get out of my talk? (The answer should not be that they will learn that you, the speaker, understand some algebraic geometry.)

Now this might be a difficult thing to do if you don't have a very clear idea of what the audience might find interesting. In that case, it is perfectly legitimate to focus on pay-off for the other grad students in the audience. You can be assured that the general mathematician's knowledge of algebraic geometry is not substantially larger than the average advanced grad student's. Hopefully, you do have some idea about what your fellow grad students might find interesting. If not, just ask them.

Finally, everyone understands that 20 minutes is not a lot of time, so just pick some small subtopic that you can reasonably say something about in 20 minutes. Less is more. If it were me giving the talk, unless I had better ideas I would probably pick some theorem from the book that I thought the audience might find interesting, illustrate it with some examples, and say a few words about how in fits into a larger picture. That's just one possible template for the talk. The most important thing is to give honest and serious thought to making sure that the audience gets something out of your talk, and to reflect afterwards on whether you have succeeded in this.

Good luck!

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    And be prepared for the following experience: no matter how narrow a topic you think you've chosen, it will turn out that 20 minutes isn't enough time to cover it :)
    – chepner
    Jan 5, 2022 at 21:46
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    This comment is spot on. I would suggest some small issue that is simple, but explored in much more depth in algebraic geometry than is perhaps the case for other branches of math. For instance, just kick around the idea of what a zero is in algebraic geometry. Everybody understands the concept for basic polynomials, but in algebraic geometry, the concept is more intricate. So that topic is something they can already relate to, and something which you blow their mind about. Jan 6, 2022 at 7:20
  • Note that your talk doesn't need to necessarily have to end with some grand proof or theorem (and probably shouldn't due to time constraints). It could honestly be just a series of questions, and then musings about the implications of exploring some problem from a different world view. You can then subtly drop in comments such as, "Actually, we can show this to be true (or false), but that is outside the scope of this talk". 20 minutes is not enough to explain some very complicated topic with proofs. Just help them to see the world through your world view. That will be quite interesting. Jan 6, 2022 at 7:25
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    Agree with all of these comments. One thing I’d add is that if there’s a topic that gets you personally excited or one you find very interesting and you think the audience may find interesting too, that may be a good candidate. A lot of great talks are great not only because they cover a topic useful and relevant to the audience, but also because the speaker has a deep personal interest or connection to the topic and discusses it in an engaging way.
    – Greenstick
    Jan 6, 2022 at 21:29

I suggest that you give the talk you would like to hear if you hadn't yet read the book. You can even tell the audience that's how you decided what to say.

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    That's very good advice. I'd say that you can even assume that your past self (who hasn't read the book) is your audience.
    – Dirk
    Jan 6, 2022 at 21:05
  • Yep, think of it as a TED Talk, where you go into brief detail about 1-2 things, but what you really want is to get people interested in the idea/topic so they do their own research or read the book. You aren't there to give a comprehensive synopsis of everything. Jan 7, 2022 at 20:22
  • @OP: Out of curiosity, do you mind telling me what this book was about? Was there anything especially interesting in it? (You don't have to reply to this comment; I just figure having a real person asking you questions might make it easier to follow Ethan's advice)
    – Brian
    Jan 7, 2022 at 22:11

Rather than trying to summarize the book, or the general subject, in a ridiculously short time, (which would be pretty boring to essentially everyone), choose a specific topic/example that is understandable to any math grad student, and show how (perhaps fancier parts of) "alg geom" address that issue. That is, new, high-end things are not just self-referential, but do address pre-existing issue. :)

Leave out real proofs.

Don't give "definitions"... but, rather, give vague-but-generally-helpful descriptions. (Seriously, no one is interested in "definitions", unless there's an issue of higher precision, and definitions would kill an awfully large fraction of 20 minutes... and no one who didn't already know them would assimilate them, etc.)

Perhaps think of it as an extended "elevator pitch"... :)


Contrary to popular belief, the only reason to give a talk is to make the audience feel better than when they walked in. That is it. The speaker wants the audience to feel better than when they walked in.

One surefire way to make your audience feel better is for them to learn something (i.e. ONE THING). A second surefire way is for the audience to 'catch' your contagious enthusiasm for the subject. Do one or both of these things and your talk will succeed.

You arent going to revolutionize algebraic geometry in your talk. Drop the notion of 'complexity equals interesting' right now. The primary pitfall that you will fall into in this scenario is presenting at an advanced/expert level. DO NOT FALL INTO THIS TRAP. If even one audience member checks out of your talk you have failed. There is never any excuse for losing even a single audience member. You are speaking to the dumbest person in the room, never the smartest. EVER.

Take your draft of your current talk and waterdown the information content by about 95%... yes, really. Eliminate all equations that are not absolutely necessary. There should not be more than one or two equations on the slides, max. You are here to give a good talk, period. Good talks do not have equations on slides. Anyone that argues that gives bad talks consistently and doesnt know it. Replace those equations with simple, pretty pictures and diagrams. People love and understand simple pictures and diagrams. If it is complex enough to require a three sentence explanation, eliminate it.

Each slide should contain one-and only one- point. Your talk is on one-and only one- topic. One simple topic, one simple slide. Do not be afraid to repeat yourself. Do not be afraid to repeat yourself. Repeat the same point with different language, leading to a crescendo.

Use the simplest possible language. This talk should make sense to anyone that listens, regardless of background. it is incumbant on the speaker for the audience to listen and understand. Understanding is not the responsibility of the audience.

Start slide 1 with something that you are enthusiastic about, and end with effectively the same message. Speakers tend to think of talks as continuous collections of slides, whereas audience members tend to consume slides individually (i.e. each slide is actually its own mini talk).

Try to emulate a circus, and not your daily lectures. Talks are show business, not research. Do not sacrifice entertainment for factual accuracy, yes really. Let the details or small print come out in questions. There will be time for questions because you are going to ...

END EARLY! anticipate filling half the allotted time, yes really! No one in real life will fault you for ending early. It will never happen at this stage of your career. this is not 8th grade speech class!

  • It reads like I met my talk buddy ^^
    – Ben
    Jan 6, 2022 at 16:03
  • I feel you, I've also been there. Since I'm doing it my resp. your way I receive a lot of positive feedback, be it people from maths, physics, engineering, or even finance and business. It's much better to present few and easy-to-grasp content instead of a lot and/or even difficult. From my feeling, even difficult content can be transported such that people understand it without bending their mind. But then you have to think about how to do that and I guess many people simply don't care whether other people understand them. Sometimes it feels like they think it's a good sign when they don't..
    – Ben
    Jan 7, 2022 at 7:42
  • i think most speakers just think its the audiences job to understand. the speaker can sound as smart, complicated and sophisticated as they want during Q&A.
    – user150207
    Jan 7, 2022 at 14:12

The answers posted so far have avoided suggesting a topic. I'll suggest one: explain what projective space is and why it's such a big deal in algebraic geometry.

The elementary definition of projective space (n-tuples up to nonzero scaling) does not suggest why the concept is so important. People who have not done anything with algebraic geometry might wonder what the big deal is about projective space when "affine space" looks to the novice like a more natural geometric thing to use. So after explaining the definition, discuss some nice properties of projective space at a level that they can understand.

Show them the real and complex projective lines are a circle and sphere, so they will understand when you say projective space is compact while affine space isn't. That is a big, big deal. All mathematicians understand that compact things are nicer. Anyone who has taken complex analysis knows that for rational functions it's really good to think about them as mappings on the Riemann sphere and not just on the complex plane. In algebraic geometry, projective space is a very useful compactification of affine space. Compare with something at the undergraduate level: closed bounded intervals in real analysis have far more useful properties than open (bounded) intervals, and that's due largely to their compactness.

For some concrete things to show about the projective plane (both geometrically and algebaically), see my answer to an old MO question on a similar topic here. That has too many suggestions to fit in 20 minutes, so don't try to do all of that. I once gave a 50-minute undergraduate talk whose only goal was to illustrate how asymptotes to a curve are revealed in the projective plane to be tangent lines to a missing point on the curve "at infinity".

Don't forget to give some practice talks in advance in front of some actual students so that you know it can fit in 20 minutes and you get feedback. Give the first practice talk to yourself first since it will probably be a disaster (time management). And expect in the actual talk to lose the last 5 minutes of material, so give yourself several exit ramps in the talk. Do not leave the only payoff to minute 19. Have multiple payoffs sprinkled throughout your talk.

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    Yes, apart from "why projective space?", the point of having many good "exit points" toward the end, when you find yourself (inevitably) going almost overtime, is essential. Do not go overtime more than some seconds. No matter what you think, it will not be greeted happily (unless you've something truly amazing to present...) So you need to prepare several "graceful exit points", so that when you discover that you'll be over-time, you can finish nicely, rather than "just stopping". Again: "exit points". Jan 6, 2022 at 1:03
  • @paulgarrett What about ending early?
    – cgb5436
    Jan 8, 2022 at 3:46
  • @cgb5436 twenty minutes is truly almost no time at all. It would look quite bad to end a 20-minute talk after 10 minutes. But ending at 18 or 19 minutes if that is a natural stopping point would be fine. You did not ask the original question, so what is it you really want to know about ending early?
    – KCd
    Jan 8, 2022 at 3:56
  • I was actually thinking about ending early in the more traditional 50-minute setting (5 minutes early or so).
    – cgb5436
    Jan 8, 2022 at 4:03
  • @cgb5436 why are you keen on trying to end a 50-minute talk 5 minutes early? I have attended many 50-minute talks, and of course it can be annoying when people go over time, but it is really rare that someone would end 5 minutes early. It's a noticeable amount of time.
    – KCd
    Jan 8, 2022 at 4:42

If you have sufficient insight for it, I'd suggest that you talk about (a) the essence of the field itself, not the book, (b) how it is related to other math fields, and (c) how it differs.

Applications of algebraic geometry to other fields is part of (b). One or two key theorems would be under (a), but insight is probably better than theorem statements.

End with some observations about why students of math might want to consider studying this field. Alternatively a few words about future directions as much as you can predict them.

The above is based on your statement that the audience will be general in nature, so you need to say things of wider interest than just the technicalities.

And, perhaps, developing an outline like this will aid your own insights into he field.


You might consider reading some reviews of the book --- MR, Zbl, AMS Notices, Bull. AMS, EMS book reviews, various lesser known journals that publish reviews (example), googling the title of the book, etc. --- and see if anything especially praise-worthy or critical catches you eye and suggests something that you could discuss.


For such talks I usually refer to myself: What are the main points I learnt through the entire book? Then I will consider whether they fit to the audience and also in time regarding broadness and deepness.

I also often thought like "Well, all others probably know at least half of the book so I should focus on the advanced stuff so I can also impress them" but, no. My feeling is, the older the people are, means the more experienced, the more they prefer the concepts and ideas instead of hard proofs and technical details. You can hardly impress them with your roughly gained knowledge but you can stick out the main and most interesting patterns, so everyone will learn something and might consider it for their own researches to dig deeper.

Despite that, the talk might also be much easier when you look at the content more from a higher level and the more fun the talk will be.

  • +1 This. When you should give a talk to an audience broadly similar to you, simply ask yourself what you would find interesting to be told about that book. Then transform that topic into a short narrative with motivation, some (technical) substance and a broadly understandable summary.
    – ojdo
    Jan 6, 2022 at 13:22

20 minutes is a good time to address one interesting specific point.

In such cases, I usually go the funnel way: a very wide overview in 2 minutes to place the topic of the book (how wide depends on the audience - if they are all mathematicians (but not algebraic geometers) then explaining what geometry is is not useful. Stating that the topic is in geometry is.

You then would quickly dive into one interesting topic.

The topic should be self-contained ("you have three doors ... should you change?"), interesting on its own (maybe something counter-intuitive), or groundbreaking.

You are then left with 12 to 15 minutes to explain the exact problem and its solutions - and possibly the 2 interesting points of the proof.

20 minutes is very good timing to train people to choose wisely what they want to say. When you have no experience, you are either at the first paragraph after 20 minutes (and your audience is gone), or you have wrapped up everything in 7 minutes (and your audience is still digesting your second sentence).


  • practice several times alone, in front of the computer and your presentation. This will give you a rough overview of the timing
  • then record yourself two or three times and review the key points. You will find the "euuuuhhh..." parts where you forgot what to say, the "I need to pee" parts where you make a small dance because you are stressed and the "I was not aware of my arms" part where you need to do something with your hands.
  • and finally have someone suffer though two or three presentations of yours, some good friends who will tell you what they understood or not

Also, speak louder. But do not shout either (except when you want to startle the audience, always a good thing). Do not hesitate to address the audience.

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