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Dylan, this is a very good question, and you got really excellent advice. I have received emails from (Italian) undergraduates and graduate students asking me how to solve random exercises; furthermore, it was obvious to me that they were not just emailing me, but several other people. They did not even write my name in full, their emails would start with something like "Hello!". These are really irritating; I write to them to point out how inappropriate their behavior is, but they never seem to get it. The last one (who was completely unknown to me) replied that "being a teacher is a vocation", reproaching me for not being very happy help her solve her linear algebra problem.

Now, obviously you would not do this. Let's assume you are writing to me. I always try reply to thoughtful questions related to my work, or to something I wrote, or to something I am expert in (as an example, don't email me with questions about group theory: if they are hard I won't be able to answer, if they are easy I will seriously wonder why you can't ask one or your teachers, or someone else closer to you). I make exceptions for students from the third world.

Introducing yourself is good, but don't make it too long. The way you have been doing it is fine. Try not to be long-winded, but give all the essential details (of course, this is not easy for someone who is not experienced, so you will get plenty of leeway from me). Also, give me some idea of your background, so that I know what kind of knowledge I can assume.

If your question is very easy and not very interesting to me, expect a concise answer: if after thinking about it you still don't get it, you should feel free to write back, explaining to me more precisely what your problem is (I mean, simply repeating the question won't do, but something like "I tried what you suggested, I can see that $X \to Y$ is smooth, but since the map $U \to V$ is not a pullback of $X \to Y$ I don't see why this implies that $U \to V$ is smooth" will work). If you manage to ask an interesting questions, so much the better, and you will get a more extensive answer, and impress me as a bonus (you never know when having impressed someone might be useful).

I very strongly advise thanking afterwards. It is polite, and creates good will. I find the idea of spending time writing an answer that is not even acknowledged very unpleasant. If you don't thank me, you are much less likely to get a reply the second time.

I always sign with my first name, and don't mind it all if you use in subsequent emails (as a matter, I prefer it, but if you feel uneasy doing this then don't, that's fine too).

Don't be shy. Spend time thinking about what you are writing, but keep in mind that you won't be held to the standards of professionals mathematicians. An in In any case, the worst that can happen is that you don't get a reply.

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Dylan, this is a very good question, and you got really excellent advice. I have received emails from (Italian) undergraduates and graduate students asking me how to solve random exercises; furthermore, it was obvious to me that they were not just emailing me, but several other people. They did not even write my name in full, their emails would start with something like "Hello!". These are really irritating; I write to them to point out how inappropriate their behavior is, but they never seem to get it. The last one (who was completely unknown to me) replied that "being a teacher is a vocation", reproaching me for not being very happy help her solve her linear algebra problem.

Now, obviously you would not do this. Let's assume you are writing to me. I always try reply to thoughtful questions related to my work, or to something I wrote, or to something I am expert in (as an example, don't email me with questions about group theory: if they are hard I won't be able to answer, if they are easy I will seriously wonder why you can't ask one or your teachers, or someone else closer to you). I make exceptions for students from the third world.

Introducing yourself is good, but don't make it too long. The way you have been doing it is fine. Try not to be long-winded, but give all the essential details (of course, this is not easy for someone who is not experienced, so you will get plenty of leeway from me). Also, give me some idea of your background, so that I know what kind of knowledge I can assume.

If your question is very easy and not very interesting to me, expect a concise answer: if after thinking about it you still don't get it, you should feel free to write back, explaining to me more precisely what your problem is (I mean, simply repeating the question won't do, but something like "I tried what you suggested, I can see that $X \to Y$ is smooth, but since the map $U \to V$ is not a pullback of $X \to Y$ I don't see why this implies that $U \to V$ is smooth" will work). If you manage to ask an interesting questions, so much the better, and you will get a more extensive answer, and impress me as a bonus (you never know when having impressed someone might be useful).

I very strongly advise thanking afterwards. It is polite, and creates good will. I find the idea of spending time writing an answer that is not even acknowledged very unpleasant. If you don't thank me, you are much less likely to get a reply the second time.

I always sign with my first name, and don't mind it all if you use in subsequent emails (as a matter, I prefer it, but if you feel uneasy doing this then don't, that's fine too).

Don't be shy. Spend time thinking about what you are writing, but keep in mind that you won't be held to the standards of professionals mathematicians. An in case, the worst that can happen is that you don't get a reply.