How does one give a mathematical talk?

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 (since closed, but still available on archive.org), 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.

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.

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?

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.