I'm looking for examples of four kinds of things:
- Material that is usually covered in standard undergraduate mathematics courses and/or in first-year graduate work (or tested in qualifying examinations) but that most mathematicians aren't really expected to know/remember: Some things that come to my mind are Sylow's theorems and their applications (for mathematicians outside of group theory and geometric group theory) and point set topology (except perhaps logicians and some algebraic geometers). If there are other examples of this kind of stuff, why is it taught in undergraduate courses? I can think of three explanations: (a) it is useful to learn (either the content or the techniques) at least once, even if people forget; (b) it is so important for people who go into that area of mathematics that it's worth subjecting everyone else to it; (c) inertia.
- Material that is not taught or covered in undergraduate courses and/or in most first-year graduate work, but that professional mathematicians across multiple specialties are supposed to be comfortable with. Things that might fit the bill (but I'm not sure) are various techniques in combinatorics and elementary number theory, and ideas from category theory. But I'm not really sure.
- On a related note to (1), mathematical skills that undergraduates get good at while studying the courses but that most of them forget even if they become mathematicians. Examples include all the tricks and techniques for integration, Sylow's theorem tricks.
- In contrast to (3), skills that people get better at in general as they do more and more mathematics. This probably includes things like a better understanding of quotients, asymptotic behavior, universal properties, product spaces, multiple layers of abstraction (like a norm on a space of operators on a space of linear functionals on a space of functions on a topological space, or one of those typical things in category theory).
All the things above are guesses and I'm curious to hear what items others have in mind and whether people think there exists any notable divide or difference of the kind I've suggested above between what undergraduates learn/get good at and what mathematicians are expected to be good at.
Best Answer
One category of mathematical result that belongs to 1 is statements that you need to know are true and that have complicated proofs. Obviously some such proofs are worth knowing because they will help you find other, similar proofs. But not all of them fall into that category. For example, almost all mathematicians can get by just knowing that it is possible to construct a complete ordered field. And perhaps a more important example: many mathematicians use Lebesgue measure, but all most mathematicians need to know is a few basic facts about it, and not the full details of the construction and proof that it works. Another result I remember my undergraduate lecturer more or less explicitly apologizing for was the simplicial approximation theorem, which I remember disliking intensely.
Why do we teach results like this? One reason is that when we teach we are not just equipping people with the tools they need for research, but also demonstrating that we can build up the edifice of mathematics from just a few basic axioms. One can argue about whether we really do this, but I think we tend to do enough to convince any reasonable person that it can in principle be done. If we were to start leaving lots of gaps (there's this thing called Lebesgue measure ... it has the following properties ... it can be shown that these properties are consistent but the proof is tedious and I'll omit it) then this valuable aspect of a mathematics course would be in danger of being lost.