Pro Mathematicians: how familiar have you remained with the material covered throughout your undergrad? (Outside your field)

Outside of your field of research / application, how much of your undergrad education have you retained?

Thought experiment: How would you fare today if handed old exams from your introductory topology / number theory / differential equations / whatever classes? No studying or preparation.

I've always wondered how much of this material one should expect to have internalized, and how much is acceptable to forget over time and need a reference aid.

Solution 1:

Professional mathematicians are not always professors. There are many working for government-based research labs, other government agencies, private research firms, and industry.

In these cases, in my experience many of them -- because they do not teach as a regular activity -- do not retain many of the details, but they understand all the concepts.

Many of these folk might not remember things like trigonometric integrals, or they might not be able to recall the proper way to solve certain classes of ordinary differential equations, but they certainly understand the math -- they just might not be able to recall it off-the-cuff.

When you work in industry or government, you have ample access to resources. For instance (just to use an off-the-cuff example), if I see $\int \arctan x\ dx$, I don't know offhand what that is. But I know it's something, and I know how to find it, and I can fully understand all of the details on how to compute it. So if it comes up in my work, I can recognize that and go from there. Memorizing it isn't particularly useful for me. I haven't had to compute the antiderivative in 15 years. But if I needed to know how to do it, it would take me less than 5 minutes to figure out.

Solution 2:

Very basic stuff, like linear algebra, basic topology, elementary number theory, elementary real analysis, finite group theory, etc comes up often enough in research, and the kind of problems you would see on undergrad exams are simple enough that most mathematicians would do fine on those tests, even if they haven't taught those classes recently (which they probably have).

I think that in more advanced undergrad and early graduate classes the situation would be a little different, the material isn't so easy and those things might not come up as often if you research outside whatever area. Plus people usually only teach advanced courses in topics that are close to their research area.