TITLE: How does one give a mathematical talk?
QUESTION [56 upvotes]: 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

REPLY [18 votes]: 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?