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.