I hope no one will object my raising this question from the dead...
One point which has been alluded to by Tracer Tong but which is worth emphasizing is that it is sometimes very difficult to justify the usefulness of a fundamental concept without starting a whole new book. Just saying "This gets very important later on" may satisfy the lecturer/writer who knows what he is talking about but will leave the student with an aftertaste of argument by authority.
This happens most often with exercises : it is very tempting for the author to take an example or a theorem from a more advanced corner of his subject and strip it down of its fancy apparel.
I'll list a few examples of mathematical concepts I encountered in this way "before their times" and came out with the first impression that those were silly and unmotivated - and changed my mind when I learned about them in a more thorough manner :
- Hyperbolic geometry (!!)
- p-adic numbers (!!!)
- Dirichlet series
- Milnor K-theory
I don't know the best option here... It is nice to see glimpses of more exciting subjects, but sometimes it is more a way to satisfy the (quite natural) inclination of the teacher for what lays further down the road.
