# How does one give a mathematical talk?

Sometime tomorrow morning I will be presenting a mathematics talk on something related to commutative algebra. The people present there will probably be two mathematicians (an algebraic geometer and a complex analyst) and some friends of mine.

Now this is the first time in my life giving a mathematics talk; it will not be a "general" talk aimed at the public but will be more technical. It will involve technical terms (if that's what one calls it) like quotient rings and localisation. As this is my first time I am obviously a little worried! I have read Halmos' advice here but I feel it is more for a talk aimed at the general public.

Several things bug me, one of them being how much detail in the proofs does one put in a talk? I am thinking obviously one does not check that maps are well - defined on the board but just says "one can check so and so is well-defined".

Furthermore, what about the speed that one writes? I am comfortable writing on the board and some people tell me I speak and write too fast. Obviously that is a problem and I need to slow down, but also I don't want to be talking too slow to bore the audience and seem to be out of passion. What is a good indicator of how "fast" or "slow" should one give a talk?

Besides, is there a way that one should "act when up on stage"? By that I mean so called "socially acceptable" do's and don'ts. I think any advice given from those who have "been there done that" would be useful for future wannabe mathematicians like me.

Thanks

Edit: Since many people have said it is difficult to give advice not knowing the audience, for the moment the audience will be a complex analyst, an algebraic geometer, one person who has just completed honours in orbifold theory, another friend in third year taking courses in measure theory,galois theory and differential geometry, and lastly a PhD student in operator/ $C^{\ast}$ - algebras

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Pause often enough to give people a chance to absorb what you’ve said. Move around enough so that everyone can see what you’ve written. (Ideally, avoid standing in front of what you’re writing even while you write it; this is quite possible, though not everyone has the knack. I always alternated between writing ahead of me and writing behind me, to open sight-lines from opposite sides of the room.) –  Brian M. Scott May 31 '12 at 12:03
Prepare a talk for the time you have, then cut one-third of it. It really is amazing how much time is needed to say anything. –  Gunnar Magnusson May 31 '12 at 12:05
@BenjaminLim If there are only two mathematicians in the audience, I don't see why you would want to get much more technical than a "general talk". Pick the most important proof (or two) and when the machinery comes in, do your best to give them a feel for how it works (without demanding the details). Never overestimate your audience's ability to keep up. Also, do not forget pants, if they are appropriate in your region. –  rschwieb May 31 '12 at 12:06
@BenjaminLim To your question about "talk not rigorous enough": NO. You may sacrifice lots of rigor in the interests of getting the good ideas across. (But do not teach them wrong stuff.) If you will be using the board, then the best advice has already been given: avoid talking to the board, avoid sluggishly speaking what you are currently writing. If I'm going to write something important on the board, I usually first say what I'm going to write, write a little, pause, review the idea aloud, finish writing, then go over it one final time aloud. –  rschwieb May 31 '12 at 12:18
@rschwieb "Also, do not forget pants, if they are appropriate in your region." I have a friend who forgot his pants on the day of his thesis defense. He'd bicycled to campus, and successfully remembered to pack all the rest of his suit.... –  RBerteig May 31 '12 at 18:19

Here are some miscellaneous tips. They are not of the "don't look at your shoes, speak clearly, etc." variety, but rather they assume that you want to give a really great talk, rather than just a bearable one.

1. Do not overrun.

2. Do not overrun!

3. Practice your talk in front of an empty room, beginning to end. Every "umm" will echo back at you so loud, that you will really want to weed those out. You will likely need to practice at least twice, taking care of tips 1. and 2. When you practice, you need to pretend that you really have people sitting there and taking notes. In particular, you will sometimes need to just pause and stare at the empty room, waiting for your imaginary audience to finish writing.

4. If you want to give a really interesting talk, it is not enough to know your stuff. You should try to tell your audience a story. So before you write the talk, decide what your story is about, then structure your talk around this. This involves deeper understanding of the topics than just understanding every step in every proof that you will be presenting. Once you know what your story is, you will likely also know, which proofs to leave out or to only sketch, where to give examples, etc.

5. Involve your audience, but ask only the most basic questions. Do not create uncomfortable silence (so don't wait for the answer for too long), but do not answer your questions too quickly either (you don't want to beat one of your audience members to it by half a second - it feels pretty bad to them).

6. Do not overrun!!

7. Since this is your first talk, it is less important to make it perfect, than to learn from it. I have written up some of my strategies to get useful feedback after my talks on another SE site, you might find it useful.

8. Enjoy the experience. Enjoy interacting with the audience. Don't forget that you are talking about something that you find really fun! But also don't forget that your audience members might be less enthusiastic than you, or slower on the take.

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A lot of people have advised me (on this page here) not to drone with "umm" and stuff. Why is this the case? Also, the motivation for what I may be talking about comes from projective varieties and things like $\textrm{Proj} R$. I have not studied algebraic geometry, so I will have to really try to provide a lot of motivation from algebra. Already I can think of 2 or 3 reasons as to why we want to localise rings, so perhaps motivation from just the commutative algebra is enough .... –  fpqc May 31 '12 at 14:33
To me, "umm" just sounds ugly. If you are serious about giving awesome talks, then you should see yourself as a performer. People should enjoy watching you "on stage". As for motivation, this is kind of hard to answer without knowing the exact background of your audience. Remember that motivation is something that motivates your audience to continue listening to you. From that point of view, it is always good to have motivation from as many points of view as possible. If you can give geometric or number theoretic examples, go for it. Otherwise make do with what you know. –  Alex B. May 31 '12 at 14:46
Thanks! +1 By the way for your answer. –  fpqc May 31 '12 at 14:50
(+1) But, I think there's one point that could stand to be emphasized: Do not overrun!! –  cardinal May 31 '12 at 18:26
@user02138: Going over time. –  Alex B. Jun 4 '12 at 2:10

I remember the first time I gave such talk, it was my last requirement as an undergrad (a part of the course I took in representation theory). I was stuttering the whole time and "ummm... uhhh..." was dense in my lecture.

Since then I began teaching and gave several seminar lectures. Let me give you some general advices which may be applicable here.

1. Know what you are talking about in details. The better you understand, the better you can explain. In the future you may find yourself giving a talk without preparing at all, and it may end up being pretty good - this cannot happen if you only have a vague idea about the topic. The more you "grok" it, the better you will present it to others.

2. Be interesting. This, at least for me, means that you need to try and keep active interaction with the crowd (which is relatively easy if the crowd is small). Try and speak in a non-monotonic voice, a constant tone and speed is a good way to put people to sleep and despair. Try to insert a joke or two when appropriate.

3. It's fine not to prove everything. I learned this from a seminar lecture given by Stefan Geschke a year ago. He proved some of the theorems but he skipped some of the proofs, instead talking about the idea behind the proof or what can be done with the theorem.

I remember coming out of the lecture wide awake and with a good sense of understanding the general idea. In contrast, some talks I attend to are filled with proofs of every single detail and lemma, and I get lost. This advice is particularly useful if the crowd already knows some of the things you are going to say. You can cite them and keep forward.

4. Don't overdo 3. You still have to prove something, otherwise people feel that you only gave some vague framework - not a talk about a mathematical concept. If you aim to prove some big theorem, write it on one side of the board along with two or three lemmas you will need, then work on the other side. This will allow people who zone-out to tune back into the talk.

5. Relax and try to have fun.

Regardless to all of the above, I doubt most people give good talks the first few times they give talks. So just remember to take it easy, it gets better with time.

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This is great advice! –  Eugene May 31 '12 at 12:29
+1 Asaf great words of advice! –  fpqc May 31 '12 at 12:30
@BenjaminLim What is the talk going to be about, if I might ask? –  rschwieb May 31 '12 at 12:33
@BenjaminLim Just one more thing to add to this. Be ready to answer questions. Some audience members will ask for clarification on certain issues and showing that you know your stuff down pat is important. –  Eugene May 31 '12 at 12:36
Related to what Eugene said, don't be afraid to not know the answer to a question - it's going to happen sooner or later. One thing I've learned is that it's really easy to ask an incredibly difficult question! –  Jason DeVito May 31 '12 at 13:02

A lot of good advice has already been given. However, there is a key question which I'm surprised that no one has yet asked:

WHY are you giving this talk?

Every mathematical talk has a purpose, and there are many different purposes for a mathematical talk. For instance:

1) The job talk: you are being (explicitly or implicitly) screened for an academic job, and the goal of your talk is to make the listeners want to hire you.

(As a good friend and colleague of mine likes to point out, in some sense every talk that you travel to a new place to deliver is a job talk in that if you do very well you will advance your career and if you do very badly your career may suffer.)

2) The research seminar talk: In this type of talk the purpose is to convey something about your own recent work to an audience of relative experts. Note that this is formally similar to a job talk (and see the above parenthetical comment), but if it is really not a job talk -- and, in particular, if it's a talk you're giving to your colleagues / peers / students -- it has a quite different purpose: to inform rather than impress.

3) The colloquium talk: This is somewhat similar to 2) but with a general mathematical audience. (Note that the phrase "general mathematical audience" actually carries relatively little meaning: if you are asked to give such a talk, you should find out more exactly what it means! Sometimes colloquia are attended primarily by research mathematicians with a particular interest in your field, and sometimes they are attended primarily by undergraduates who may or may not be math majors.) The purpose of a colloquium talk is somewhere between information and entertainment. Although you are expected to talk about your research, this can sometimes be in a very general way: for instance, you can give an excellent colloquium talk in which you do not explicitly state a theorem of your own (but it takes some guts and confidence in yourself to do so).

4) The learning seminar talk: This is a talk that you give as a participant in a seminar. Generally you have chosen, or been assigned, some specific paper or part of a paper and your job is to present as much of this as possible in the time allotted, starting from an overview but often including key technical details.

5) The student project talk: This is a talk that a student gives in the context of a particular course. Often, but not always, it is given "in class" and only attended by other members of the class and the instructor. Often there will be an accompanying written paper that you are summarizing. This is similar to 4) but probably at a lower level.

6) The expository talk: This is a talk with the goal of teaching the audience some material which is (usually) not due to you. Really it is like teaching a course except it all takes place in one sitting (or maybe a small series of sittings), which of course makes it challenging. Generally you have some specific reason to do this, e.g. it functions as background / prelude to some other activity.

This is probably not a complete list, and if anyone wants to suggest any additions, please do so.

So...what kind of talk will you be giving? Any of the above? Did someone ask you to give this talk? If so, why?

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To echo Pete's answer: indeed, there is a purpose to any talk, and that should determine most aspects. Sometimes people slip into thinking that there is an "abstract" notion of "talk" that is independent of context or purpose, but that leads to self-sabotage. The purpose, the audience, the expectations, the goals, ... must be considered and responded-to. These and the time constraint usually (nearly-) completely determine what one should do. –  paul garrett May 31 '12 at 18:32

I have been giving long talks professionally for about twenty years, with great success, and the best advice I can give you or anyone is to give at least one practice version of the talk first. Find someone to give it to if you can, or if not then give it to your dog, or to an empty room. Actually say the words aloud. This will give you a better sense of how long your talk is than anything else you can do. But equally important, you will discover all sorts of things that seemed clear and straightforward in your head that somehow turn into garble when they come out of your mouth. The practice talk is a chance to fix this.

One problem that many beginning (and experienced) speakers have is the droning problem that other posters in this thread have alluded to. This is not just the problem of pausing and saying "ummmm." (Or its British counterpart "Errrr"; in Norwegian it is "Øøøø".) It's that a lot of people who are perfectly good speakers around a café table or in a bar suddenly lose all their intonation, inflection, and pacing when they get up in front of an audience, and speak in a robotic monotone in which all the syllables are the same length. One strategy for avoiding this is to make a conscious effort to pretend that you are sitting at a bar with these people after a round of drinks, and try to speak as if that were the case. Another strategy is to pick one person in the audience who seems friendly and pretend that you are giving the talk just for them.

I sympathize with your problem of talking too quickly. I do this too. If you can't talk more slowly, there are two things you can try that might be almost as good. You can insert a long pause after every sentence or two. (I once gave a talk in Taiwan that went very well, in part because I had to pause after every sentence or two to let the translator repeat my remark in Chinese.) Or you can repeat everything two or three times, which is not as awful as it sounds.

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There is one problem with talking "to one person", you feel that you ignore the rest of the audience and it can bother you as a speaker. I know that sometimes when talking to a class full of people I may feel that I focus on one person and it makes me feel a bit uncomfortable. –  Asaf Karagila May 31 '12 at 16:40
Yes, it's easy to go too far with that, particularly in a small room. But I think it might be better to give the talk to one person in the room than to give it to nobody in the room! –  MJD May 31 '12 at 16:42
(+1) I've found that practicing once helps, but practicing 3-4 times is usually necessary to work most of the verbal kinks out. Stand in front of a mirror, point at your laptop screen as if it were the projector. The mirror will help catch any unusual mannerisms or other quirks that need polishing. –  cardinal May 31 '12 at 18:29
Talking to one person is better than talking to nobody, but it usually better to look at everyone. Some people feel compelled to test the speaker with questions if they feel ignored. Also, accoustics often force you to speak differently than at the coffee table. Stage actors spend lots of time on their voice. –  Michael Greinecker Jun 1 '12 at 8:01

When I did my first math talk, I read all the answers here.