Do Math Professors still do textbook problems to learn subjects foreign to them?

Question

This question is very subjective and is of no technical detail, however since MSE is a place for working mathematicians, I thought I'd ask here to get the best answers!

As a prospective graduate student in mathematics, I have come to know the mountain of knowledge is quite tall and there is much to learn, even past graduate school.

However, I would like to think that if my goals of becoming a professor come true and I become a specialist in an area of math, that even though I specialize in one area, that I would still love to continue learning other areas of math.

As a working mathematician in a certain area, do you ever find yourself as a working analyst picking up Hartshorne and finishing it 3 months? Do you as a differential geometer find yourself interested in Diophantine equations and pick up a graduate number theory book and go through the exercises?

If this is the case, do you normally read textbooks as a fulltime professor and go through exercises just as in university?

Or, is it unrealistic to spend time on new material when this time could be spent on your area of research?

Answer 1

This is just an opinion, based on my own experience as a working mathematician. Let me say honestly that a professor seldom studies a subject as a student would do. In our times, publishing has become an urgence, so that we must publish as soon as possible.

Therefore, we do not usually move very far away from our expertise, unless we are the few outstanding mathematicians that can work on almost any subject. I could tell you a couple of names, maybe three. If, while working on something that we know well, we need to learn some new result from a neighboring area, we tend to study what we need. A textbook might be useful, but we can't read it from the beginning to the end.

Last year I had to learn some analytic number theory, and I picked up a few books on the subject. Today, I have already forgotten everything, since I did not study theorems deeply. I regret this, but learning a discipline like a student would require hours, days, weeks of hard work. I simply don't have so much time to study the way I did when I was a student.

Answer 2

If a mathematician needs to learn a subject and if he or she has a textbook then of course he or she does the exercises! Since he or she has presumably become a better problem solver than the typical student many of the exercises will be quite trivial, but the ones he or she doesn't get immediately are the important and useful ones.

He or she is lucky if a textbook on the subject even exists. It will often happen that there is no text - in such a case he or she, at least in my experience, will make up exercises to do. (Often without explicit thinking "hmm, what are some good exercises?" - when one learns a theorem it leads naturally to questions that one naturally tries to answer.)

I recall shortly after getting my degree: I already "knew" basic analysis, but inspired by something one of my professors said once I went through Rudin Real and Complex Analysis and did almost all of the exercises in most of the chapters. Was somewhat less stupid after doing that.

The last few years I've been learning a little (linear) algebra that I should have learned in school but didn't. I find the only things I really understand are the things I've worked out myself, so even thought plenty of books exist I've ignored them and taken things like the Jordan Canonical Form or the Cayley-Hamilton theorem as exercises... (I've decided that the key to all wisdom is the fact that $K[x]$ is a PID.)