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Question: How does someone go about learning advanced topics in Math after they're done with their PhD?

Specific example:

  1. You've done your undergrad and masters degrees in math and learned from whatever courses they offered

  2. Then you finished your PhD in something like say...Lie Groups but in all that time, perhaps you took differential geometry but never touched anything like Symplectic Geometry or Fukaya Categories or Toric Varieties.

Those are advanced topics that are outside of what was done in your coursework and research but still, you might not be able to just learn these topics entirely on your own. At least, that's how it seems to me.

I suppose some more direct follow-up questions are in order:

  1. Is it embarrassing to have to take courses after you're finished with your PhD and to need assistance with advanced topics?

  2. Is there an expectation that you should be able to learn almost any further topics on your own without needing to in some sense 'be a student again' ?

I hope my questions are sufficiently clear. Any feedback is much appreciated.

Salazar_3854708
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    I suspect the answer depends on the person. Personally I would not really care about what others think if was to take a course. – copper.hat Jan 15 '22 at 19:28
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    Same here. I personally don't think there is anything to be embarrassed about being a student again. I am currently a postdoc and I have attended a few courses during my time at my current institution. Before the course I didn't know anything about the topic and after it I knew slightly more. That's all there is to it. – almosteverywhere Jan 15 '22 at 19:35
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    It was very common for post-docs at the schools I attended to sit in on "topics in..." courses, or in some cases teach unofficial "topics" courses, in order to learn. This was even true of full professors on occasion. – Brian Moehring Jan 15 '22 at 19:40
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    I'm still a student. I think I'm in $58^{\text{th}}$ grade, although nowadays I have to recompute my grade from first principles whenever I want to know... oh wait, it's $59^{\text{th}}$... – Lee Mosher Jan 15 '22 at 19:46
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    I'm confused with your question, since after at most a couple of years of graduate study, most of your learning occurs outside of class. But to be more specific . . . reading/scanning journal papers, working through appropriate books (rarely from start to finish), teaching classes, writing papers on things of interest to you, etc. Even writing lengthy researched "answers" in stack exchange! (example 1 and example 2 and example 3) – Dave L. Renfro Jan 15 '22 at 19:48
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    The fastest way to learn a subject, as a researcher, is to write a paper on that subject. On the other hand, if you are not working in that subject, you won't have time to learn it I suppose. – Arctic Char Jan 15 '22 at 20:10
  • @ArcticChar I gather you mean that it forces you to go through the literature and such. That is clear. But you'd likely still have need to talk with specialists, no? (and ask naive questions at sometimes) – Salazar_3854708 Jan 15 '22 at 20:15
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    If you want a specialist to talk to you, you need to bring something to the table. – Arctic Char Jan 15 '22 at 20:22
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    Another way to learn a subject is to well prepare a class and lecture on the topic rather than attend it. – Maksim Jan 15 '22 at 20:34

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