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Suppose some has done coursework in Algebra and Analysis. After that they want to pursue a PhD in commutative algebra or in Ring theory or any pure mathematics topics.

My question : In PhD programs, what type of work are PhD students doing after the coursework?

My thinking : I mean, there are lot of theorems in commutative algebra. Did PhD students modify all theorems or do they choose a particular theorem and try to modify that theorem? Also, there are lots of exercises in Atiyah's commutative algebra book. Do PhD students choose a particular/specific exercise and work on it?

Note: I'm a MSc student

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Choosing a suitable problem requires deep insight. Often a student just finishing coursework won't have that insight yet. Insight comes from very deep dives into the essence of a problem space. It is an emergent property, not easily or automatically attained.

Hence, students are often guided to a problem (or problems) by their advisor who does have this required insight.

I once had deep insight into classical real analysis and classical topology, but almost none in algebra. This seems odd, but I think it common. I entered the research stage as a good and hard working student, but didn't have sufficient insight yet to come up with a problem on my own. My advisor was a great help.

In fact, it was in working on my eventual dissertation problem that I got that insight. I don't think I could have really considered myself a mathematician until I was at the end of my research. Only at that point was I able to "bank" a number of future work problems that seemed to be important enough for further study.

Get a good and experienced advisor. Accept their guidance.


There are a few students who don't need this advice and come with a problem of their own. Sometimes that works out. But I doubt that, in math, at least, that it is the most common case.

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  • Thanks you sir @Buffy Can you tell me some reference or books .From , where i can get this problem for research/phd in pure mathematics?
    – jasmine
    Commented Oct 6, 2021 at 20:18
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    Sorry, not much help there. But I found that doing a lot of exercises/problems helped me develop that insight. Look for common threads. Look for relationships.
    – Buffy
    Commented Oct 6, 2021 at 20:31
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Typically a Ph.D. student would have had enough exposure at the graduate level to be able to choose a problem they find interesting as their thesis topic with the help of their supervisor.

It does not always have to be a novel solution. For instance, it can be a different way of proving something already proven.

They also can continue to study and take relevant coursework to their thesis topic in either reading or classroom courses.

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    Often the choice of problem is guided by the advisor. And, importantly, a new proof of an existing theorem has to be sufficiently novel that it gives insight beyond the theorem proven. That requires insight.
    – Buffy
    Commented Oct 6, 2021 at 19:52
  • That's a great point @Buffy! Commented Oct 6, 2021 at 19:55

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