(Feel free to close this if it is too vague/chatty/soft/etc, I won't be offended!)
Some very quick background. I am a visitor this year at some university, and they have very kindly organized a number of events for all "temporary" people (so postdocs and visitors such as myself mainly). Some of these events are social, but I want to talk about the "Colloquium" which we run (mostly between us though some extra people show up). I want to talk about the difficulty we have to understand each other.
Apart from being the "temporary" people, on a mathematical level we have little in common. I am deeply frustrated by how little we are able to share in the colloquium.
The first talk was about number theory, modular forms in particular. The first sentence was "as you know the Galois group is profinite, that is, compact and totally disconnected". The PDE people in the audience rolled their eyes, as you can imagine, and pretty much stopped listening after this one sentence.
We talked about this, and the next speaker decided he'd keep things very basic. He gave a talk on PDEs, and I was only able to follow about 10 minutes. Which is terrible.
When it was my turn, I tried to do something about the quadratic reciprocity law (in fact based on a MO question!). It's not for me to say how it went, but I remember being in shock at some point: I was saying "this set $G$ is in fact a group, it has $n$ elements, so if $g \in G$ we have $g^n = 1$". Somebody (a respectable specialist in probability theory) asked "why is that?" Having learned from this experience, I know that it's worth saying that it's called Lagrange theorem, and believe me, whoever you are, you've learned it within the first two years of university. It blew my mind to find out that some mathematicans have forgotten this one — but of course the things I have myself forgotten must be equally fundamental to others.
Hence my question:
Would it be useful to write down a document specifying what ALL working mathematicians can be expected to know? People giving a colloquium talk (as opposed to a seminar talk) could be required to adjust their presentation so that it is understandable by anyone knowing what's in that document.
I'm thinking of something similar to what the Word Wide Web Consortium (W3C) has achieved: a standard, a protocol.
I want to stress a difficult point: it's not only about known results, but also standard habits with notations. Let me give you an example of the things which confused me greatly during the aforementioned PDE talk. There was a map $(x, t) \mapsto f(x, t)$ and at some point the speaker wrote $\hat{f}(t)$. I was unable to decide if he meant (i) fix $t$, take $x\mapsto f(x,t)$, take the Fourier transform of that, call it $\hat{f}(t)$, it is a function; or (ii) fix $x$, consider $t\mapsto f(x, t)$, take the Fourier transform and evaluate it at $t$, call it $\hat{f}(t)$, it is a number; or (iii) something else.
I was uncannily reassured to find out that next to me, some other specialist in PDE said "that's funny, I would have written $\hat{f}(x)$ for the same thing". But then I was more confused than ever when they agreed that the notation did not matter since the meaning was obvious anyway (!). I'm not more anal than the next person, and I'm certainly no Bourbaki fanatic, but I found the need for clarification. (I tried to ask a question but they thought, again, that I was argueing against the notation, not that I was utterly confused as to its meaning.)
I would be interested in reading your thoughts about this. Of course a subsidiary question is, who would write the document, and what authority would it have?
(anecdotal stories of complete confusion during a talk can also provide comic relief, by the way)
Thanks for reading,
Pierre
EDIT: based on one answer below, I want to add the following: I'm not looking for advice on how to improve my skills in exposition, I think I'm doing fine, thank you… The suggestion I'm making, should it be efficient at all, would be rather to improve the average quality of exposition in talks. And specifically, when a speaker adresses an audience of non-specialists. Of course whenever a speaker truly cares (and I think I do, for example) s/he will be doing fine. The point is to make them care.
EDIT: turned into community wiki.
Best Answer
I really don't think any good or useful could come out of such a standard...
I sat through a plenary talk in a conference with a Fields medal recipient sitting right beside me. The speaker was very much aware of his (rather imposing!) presence---we were sitting on the front row---and, after the initial minutes one could say that he was talking to the medalist. Now, after a good 30 minutes the said medalist asks me very quietly «do you know what some-concept-or-other is? I think I am being supposed to know about it...» I remember pondering at that moment the fact that I had taught a class recently to undergrads about that and, to be honest, the incident managed to considerably increase my respect for the guy.