How much new mathematics should we learn during the PhD? Are the courses we take during the PhD really important?

Question

I was studying some papers of a really good mathematician during my masters and I got really interested in following this field during my PhD. Because of that I am now in the US doing a PhD under the advise of such a mathematician. The problem is that this University is focused in Applied Math, so I do not have many options to take courses in pure math. And the closer I can get of one of those pure math classes is a self studying course. Also I am not very fond of this math department because there are just a few of students, and all of them are in applied math, so I do not have much to discuss with them. I came here knowing all that, however once I am here I feel that the reality is harsher than I expected.

Also, as it is the natural in the US, I am a Teaching Assistant so I can fund my studies here. So I am basically paying (through work) to be in an enviroment that does not really fits me. Let alone the time that I need to put into the TAing, which takes me time from learning new math and from the research.

My research is going well, and my advisor is a great person as he is as a mathematician.

However, in my home country (Brazil) I can pursue a PhD being totally funded by the goverment in one of the top Universities of the country. And there I can also taste a lot of new math, have plenty of time to dedicate.

I am in the first semester of my PhD here in the US. But lately I have been questioning whether I made the best decidion for me. Sometimes I think about going back to my country and pursue a PhD there. I wanted the opinions of some people here; what do you think about it? Should I stay here and take a PhD degree in the US under a great mathematician, or should I invest more in my time doing and studying math in Brazil?

Thank you.

Answer 1

I'll assume your degree is in math and probably some aspect of applied math rather than pure (theoretical) math. Everyone's desires are different and some mathematicians, though I suspect pure math, are driven to study as many aspects as possible.

But the more important thing, in a degree program is to follow the guidance of the program and the faculty and study what is necessary for the degree. This is a short term thing that needn't last your whole career, but you need to get the degree to achieve that career (pure, driven, mathematician writing).

This is especially true if you have to take preliminary/comprehensive exams to earn the degree. And also especially true if you are in a research program that uses only a small subset of the huge range of math topics. And, in such programs, the required courses can be extremely important if they enable passing those pesky comps.

If you are studying under a "great" mathematician, ask them for advice in this. Ask for guidance on a path to initial success (getting the degree). Once you have that, you can "go wild" in studying what you like, though early career requirements have some limitations as well.

There is a saying among (pure) mathematicians that you don't choose math; math chooses you. If that isn't you, and you aren't driven by math, then look for an efficient path with guidance.

Since you seem to already be on a path to success, I would hesitate to suggest you give it up for a different sort of program. Get the career started, reach a point where you can set your own agenda, and then "go wild" if you like.

And: "Don't let the perfect be the enemy of the good." (Voltaire: "Perfect is the enemy of the good")

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Let me note that there is a "mathematical way of thinking". True enough, but I've also noticed over my lifetime that deep insight into one subfield (analysis, say) might provide little insight into other subfields (algebra), even though both are axiom based. The sorts of questions asked can be quite different. The implication here is that it can take a lot of time and effort to learn (i.e. gain sufficient insight into) new topics, even for experts in other topics. Get the degree then "go nuts".