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I am an applied mathematics student and recently finished my MSc.

During my second year of the bachelors I had many personal problems and was at risk of losing the entire year. To avoid this delay, I rushed the courses I was to follow and passed the exams without really learning the material. Ever since, I have the feeling that I always managed to 'swing by' with very large gaps in my mathematical toolkit. After my bachelor ended I jumped straight into the MSc without thinking much about it due to financial reasons (taking a break and starting at a later point would not have been feasible).

Throughout my master's program, I particularly got along with a professor but did not work with them for my final thesis work. After I finished my MSc the friendly professor offered me a PhD position, even though my supervisor did not. I hope not to sound boastful, but I sense that the professor in question, although a wonderful person, isn't a strong mathematician, neither pure nor applied. He lacks basic mathematical knowledge(basic set theory, measures,etc...) and even basic applications are not very clear to him. Moreover the works he has coauthored are more basic statistic tests rather than any applied mathematics, and lacks an area of expertise. The subject of the PhD is interesting as it is broad but the type of approach he wants to use to tackle it is not particularly interesting to me. Although I think I would enjoy a PhD, these reasons make me question whether this particular PhD is for me or not. On one hand, I believe I would enjoy the 'freedom' of researching topics that interest me, on the other jumping and committing to such a long project scares me. I have not been extended any other offers nor have I applied to any positions in particular. I have also been struggling to find a 'math' job that does not require a PhD. All positions available are more 'software developer' or 'data analytics' than applied mathematics.

Part of me is scared that having large gaps in my curriculum and working with someone that has the same might not be constructive for me. How important is the supervisor for my personal development as a mathematician? The other part of me wants to accept the PhD offer and see where it leads me since the supervisor and I get along well and not have to worry about the uncertainty of the future. Does taking time off after university, to figure out what to do and fill the gaps in my knowledge, preclude a future PhD and possible career in academics?

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    Should the title say Potential PhD student? I don't see that this is about the advisor. Commented May 16 at 21:46
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    @JosephDoggie It's in the third paragraph.
    – user176372
    Commented May 16 at 21:59
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    Focused questions are a lot better; we probably shouldn't have to wait to the third paragraph to start to have any idea what the title is about.
    – Bryan Krause
    Commented May 16 at 23:21
  • What sort of career are you anticipating? Commented May 17 at 0:10
  • Did the professor who offered you a PhD position have any other PhD students in the past? Or supervised a masters thesis?
    – quarague
    Commented May 17 at 7:48

3 Answers 3

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Mathematician here. Regarding gaps in your knowledge, you should know that most mathematicians feel there are things they wish they understood better. It goes with the job. As you go on, you get lots of opportunities to learn things more deeply. In your case, the PhD qualifying exams might be an opportunity to shore up those gaps. At this point in your career, you are more than capable of reading undergraduate math books on your own. Also, you might want to read up on imposter syndrome. Now, to your actual questions:

How important is the supervisor for my personal development as a mathematician?

Very important. This is someone you will work with for years. It's essential to be interested in their line of work, and aligned with their way of thinking, approaching problems, etc.

The other part of me wants to accept the PhD offer and see where it leads me since the supervisor and I get along well and not have to worry about the uncertainty of the future.

You can never get away from worrying about the uncertainty of the future. In general, it's good to be intentional about your life choices instead of letting random chance make the choices for you.

Does taking time off after university, to figure out what to do and fill the gaps in my knowledge, preclude a future PhD and possible career in academics?

No, it does not. There have been many questions here about slightly older folks going back to get a PhD and shift into academia. That said, if you don't use your math knowledge it will start to slip away, so the easiest time to start a PhD is probably now, from the point of view of preparedness.

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  • This is all good advice for multiple of the points OP mentioned in their post. But it doesn't address the title question, namely that OP is unsure whether their potential advisor is sufficiently qualified to supervise a PhD student. Unfortunately I don't have a good idea how to address that either.
    – quarague
    Commented May 17 at 7:49
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    @quarague Ok, I'll answer that. If the OP is correct (and, it's not clear to me that the OP is really in a position to judge this, as there's a lot of taste that goes into what makes good math, and the OP is still developing their taste), then the answer would be that one should not work with such an advisor. You need a strong advisor, with good ideas, a good publication record, and respect in the field. Their influence partially determines the student's success at getting a job after finishing the PhD. Commented May 17 at 12:37
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I feel like I have just lived through this on the supervisor's side.

I had a student who did not complete tasks and was generally very difficult to supervise. Eventually, I had to tell them that their behavior was unacceptable.

I found out that:

  • They did not get into the program they wanted, and I was their backup solution.
  • They consider my research second-rate.
  • They intentionally ignored my advice because they consider me a bad researcher and believe it is in their best interest not to follow my footsteps.

This student felt justified in their position because, during their time under my supervision, they were not able to publish. They were resentful towards me, believing it was my fault they had not published.

In retrospect, I understood that it was a vicious negative feedback cycle. I was unable to help them publish because they would not heed my advice, which led to them listening to me even less and provided me with even fewer opportunities to support them.

(For the record, all my other students publish in the top venues of our field of research).

Needless to say, this student is no longer under my supervision. It is impossible to supervise someone who does not respect you. The whole experience was painful and a waste of time for both of us.

Do yourself a favor and either:

a) Politely decline.

or

b) Acknowledge that you are in a position where you have a very limited view of the field, and that other people, with a broader view of the field, trust this person enough to provide them with funding to hire you. Find out what this professor does well, and learn to appreciate their work. Find out how they can support you. Once you have identified the professor's strengths, reflect on their weaknesses again, and figure out if they can still be valuable to you. Maybe even in conversation with your professor. Don't say "you are bad at X," but say "I noticed you do not focus on X, but X is something I struggle with. Would you be open to seeking collaboration with an expert in X, as I wish my PhD to include X?"

The relationship with your supervisor is essential especially during the early phases of your PhD.

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Not a mathematician. PhD is a long journey. Regardless of whether the professor has the qualifications or not, you should not start your journey with such doubts. Pauling got the structure of dna wrong — basic mistakes apparently as per wikipedia Linus Pauling. He was already well established by then. Any new work requires iterations. With your doubts, constructive iterations seem unlikely.

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