conferencesmath-communicationsoft-question
I was wondering what would be the best way to present your paper at a conference, if your paper is selected for "short communication", lasting for about 15 minutes?
Should you concentrate on the main results or the proofs?
And what should a first-time presenter be wary of?
Thanks in advance.
I'm in somewhat similar position too. My attitude is that most time people know why they are asking for some particular result so it is their headache to convert it into "public good" and to "spread the word", so to say. I have some other fish to fry (including thinking of other math. problems even from the same MO). Sometimes I like the result so much that I even present it on a seminar or a colloquium talk. However I'm just too lazy to go beyond that and to make a formal publication out of it (that always comes at a cost, not just if you submit to Monthly).
But that is about me. Your attitudes may be quite different. So, let me try to answer your formal questions now.
1) You've done the most essential part already: you communicated the information to a person who really needed it and openly declared so. True, there may be some other people who might find it interesting, but there are many other interesting things for them too, so no efficient directed communication is usually expected or even possible if you just have a funny trick to share. You can always throw a few more drops into the huge river called "mathematical literature" and there is a non-zero chance that somebody will swallow one of them when drinking his daily cup (bucket, barrel, cistern, perhaps even "pond" for some really top guys, still never the whole river or anywhere close to it) of information but the only sure way to attract reasonably wide attention to your trick is to use it to solve some known open problem.
2) No, it isn't worth the effort. But so isn't the life unless you are one of very few people who really make a noticeable change in this world for the better. Still we all are living just for the fun of it, so if you feel like writing an article, such considerations shouldn't stop you. As I said, I prefer thinking to writing any time I have the choice, but there are many people who love writing so, if you are one of them, just go ahead and write.
3) There are several contradictory ones. So, just rely on your taste and common sense and proceed by trial and error. Don't get surprised though if it turns out that the ideas of other people about what is interesting and what is not don't coincide with yours more often than not. If you really want to write for them, try to understand their point of view instead of getting angry and frustrated. Snobbish remarks of the type "this is not research level" count as "a point of view" only if the person making them can do what you've done with his little toe on the left foot alone, but there are justified explanations of why some things are not worth publishing (the most typical one is "Lemma 5.1 in this textbook covers your approach in a much more general setting"). Warning: what is new and exciting for you may be a boring routine for humanity; after all it is not easy to notice something that seven billion other people overlooked. So, you'd better have some indication of an outside interest to your idea before trying to spread it unless you want to end up in the same position as I when, being a pupil in a math. school, I decided to accurately write some complicated proof by induction of the fact that the Euler function $\varphi(n)$ is even for every $n>2$ to impress my classmates with it. The end of that story was most embarrassing, of course.
I once wrote a paper with an undergraduate that I thought was very nice. After it was accepted for publication, we found a paper not only proving our results, but going a step further. We hadn't found it previously because, similar to your situation, they used different terminology. In our case, our proofs didn't add anything new, and our results were weaker, so we withdrew the paper before it was published. That same undergraduate went on to write more papers with me, and is currently a successful graduate student. I mention this to emphasize that not publishing is not the end of the world.
If you decide to publish, you should definitely cite the work you found, and mention to the reader that their proofs are different and came first. Something like the following should be fine: "After a preprint version of this paper was made public I was made aware of [Y], which appeared first and obtained Theorems ###. As the methods this paper are completely different, we believe that the proofs still merit publication."
It is not necessarily your job to explain those differences, unless you think it would help the reader. However, it is your job to convince (yourself, the readers, and) the referee that your paper is worthwhile. So you need to deeply understand the differences, and perhaps explicitly point out what your paper adds to the literature.
I imagine that the authors of Y mentioned their paper because they have priority, and would appreciate citations to their work. After you have cited their work, mentioning their priority, you owe no more obligations to them. (Although, if they contact you again, you can treat their communications with respect!) Don't try to mind-read their intentions. Focus on your own beliefs, whether you personally think your proofs would help readers understand the topic more deeply and give them interesting techniques. If you like, you can send Y an updated version of your paper, which clearly gives them priority, and kindly ask if they have any further comments.
Best Answer
The first priority is to state your main results and explain why they are interesting (e.g. how they fit in with related work). With only 15 minutes you do not have much time to discuss proofs, but it is nice to give a brief outline of the proof of your main result and what is involved.
As a first-time presenter, I would watch out for the following:
First, as someone commented, it is crucial to not go over time. Don't try to cram too much in.
Second, watch out for the mechanical aspects of the presentation, e.g. is your print large enough to read, and is your voice loud enough to hear.
Third, strive to emphasize the most important points and not get lost in less important details. This is especially important when you only have a short time to talk.
Fourth, know your audience, so that you have some idea what you can assume is known and what you need to review.