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 welldefined”.
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∗ – algebras
Answer
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.

Do not overrun.

Do not overrun!

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.

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.

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).

Do not overrun!!

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.

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.
Attribution
Source : Link , Question Author : Community , Answer Author : Ilmari Karonen