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I've recently started a PhD in pure mathematics, and I'm about to finish my first year. One thing that I have become acutely aware of during this time is that my understanding of much of the mathematics that I learnt throughout my undergraduate and masters has fallen out of my immediate understanding or memory. It seems to me that my peers have a much greater all-round knowledge, and I'm concerned that I am considerably below standard in terms of mathematical competency. I've not directly broached this subject with my supervisor, however he assures me that I am on track and my work is going well. As far as he is concerned I'm doing fine, but I have a strong sense that my understanding of things like topology, analysis, number theory and algebra should really be at a higher level.

My question then is this: If you've completed a PhD, or are currently studying for one, have you experienced something similar? Is it common to have feelings of inadequacy in terms of mathematical ability? And finally, do you have any advice for how to overcome these feelings?

An obvious solution would be to brush up, but of course that is easier said than done when I have my research to be getting on with. In any case, I'd be interested to hear other peoples experiences with this sort of thing!

SeraPhim
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    Mathematics is so a huge and deep tree today, and more and now more new silhouettes appear for you on your way.. I'd say it's a common feeling, don't be afraid of the wild, especially if your supervisor says you're on track with your PhD. – Berci Jul 07 '20 at 13:49
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    I second the idea that this is a common feeling and that you should take heart in your supervisor's judgement. I also personally think that graduate students are often skilled at signalling a deep and broad understanding of mathematics by having good memories for jargon and stating theorems and knowing the right time to nod their heads during classes and seminars. But this doesn't mean they necessarily understand things more deeply than you within any significant margin of error. – halrankard Jul 07 '20 at 14:18
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    @halrankard I'm certainly guilty of nodding my head when the seminar speaker mentions some jargon or theorem I have a vague understanding of, and if that places me within the ranks of other graduate students then I'll happily be guilty of that. Thanks for the advice! – SeraPhim Jul 07 '20 at 14:29
  • not exactly an answer to your question, but here's something to cheer you up:) https://math.stackexchange.com/questions/237002/too-old-to-start-math – Alex Jul 07 '20 at 14:29

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That sounds like a typical impostor syndrome. Without knowing you personnally it is of course impossible to claim so with 100% certainty, but if your advisor is happy with your work so far this is a very good sign and you should try to carry on and not worry excessively.

I think having a period of stress/depression/feelings of inadequacy at some point during the PhD, especially at the end of the first year, is extremely common. Probably most of your peers feel them too or will at some point.

Albert
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  • @JCAA Yes I do, I mentioned that my supervisor is happy with my work, 6-th line down! – SeraPhim Jul 07 '20 at 13:50
  • But yes, my supervisors reassurance has helped a lot. I've yet to talk to the other PhD mathematicians in my department about this, so I don't actually know whether or not they've experienced this. Although I'm sure you're right that it is common and it's likely they have. Thanks for your response! – SeraPhim Jul 07 '20 at 13:54
  • @JCAA Such details depend on OP's location. Unless things have changed from the 80s in the US the 1st year grad students typically don't have an advisor. At Notre Dame one tenured professor was nominally a supervisor of graduate students, but IIRC he didn't do much visible work with us. Here (=Finland) the supervisor must be a tenured professor, but the advisor may be a post-doc in the old fart's team. When you get accepted to the grad school, both will be assigned. I guess there is more variation worldwide. – Jyrki Lahtonen Jul 07 '20 at 13:55
  • @JyrkiLahtonen I'm in the UK, over here we get a supervisor from the word go, along with a secondary supervisor, although their role varies depending on the project (I've barely interacted at all with my second supervisor). I'm unsure of the conditions needed to take on a PhD student, but my supervisor isn't a tenured professor, so it certainly isn't as strict as in Finland. – SeraPhim Jul 07 '20 at 13:58
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You feel the same as I did during periods of my PhD. I think this is entirely normal. I will add however, that since we don't know each other, I am basing this 'normality' on scant evidence.

It is quite normal to have these feelings around your performace as a Mathematician and how you compare yourself to other PhD students, that you yourself say are coping better than you How do you know this I wonder? If you are comparing their performance within their own field of study and you say they are more competent than you, is this really true or are you searching for a self fulfilling prophecy?

As others point out if your advisor is happy with your work so far then I wonder if it is a good idea to speak with him/her? (s)he may be able to assuage any fears you may have and may very well be able to help you.

Keep strong, this will not be easy to complete but screw your courage to the sticking place and you'll not fail!

  • Oh yeah I'm certain that I am overthinking the competency of my peers. In truth I don't have any evidence that they are any more or less competent that I am. At the moment, my worries aren't too severe to speak directly to my supervisor, although I will talk to him if they do. For now, I'm just curious if other people have experienced this as well, and what they did to get past it. In any case, thanks for you advice and encouragement, I appreciate it! – SeraPhim Jul 07 '20 at 14:09