I'm a mathematics PhD student and I have a question. I am writing my thesis, and I'm looking for a postdoc too. I would like to do research long-term, but I am not sure that I will be successful as a researcher because of the following problem.

I am very good in the research that I am doing. Everyone confirms that I am very good in what I am working on. But, I have a big problem that I can not solve simple questions. Let me explain in further details, some month ago, I wanted to use the fact that supremum attains for every upper semicontinuous function on compact space. My supervisor asked me to prove it in front of him, but I couldn't do it. These happens a lot to me. My supervisor said he did not know how to possible someone do high-level research, but he couldn't prove simple things either. I am suffering because I think everyone can prove simple things, but they can not do high-level research. Does someone have any ideas or advice for me? I want to improve this issue, because I have a final exam for my PhD. I am so scared that they maybe ask some basic questions that I do not know.

  • “supremum attains for every upper semi continuous function on compact space.“ For what it’s worth, that sounds like higher level maths than I’ve ever done at all as someone in the IT space, so it’s possible that you’ve just got Imposter Syndrome, though I’ll let someone more familiar with high-level mathematics Answer this since I’m not sure. – nick012000 Aug 26 '19 at 22:12
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    Doing things at your own pace is generally very, very different than doing them under time pressure (do it now) and in a socially pressured situation (do it while I stand here and disapprove at your lack of effortless performance) - much less both together. If you were asked to prove the same thing and bring back the answer tomorrow, or to the following weeks meeting, and you could do it, this suggests it is the implied time limit and/or the demand to do so on the spot or be judged harshly that you are having trouble with. That can be overcome or bypassed, and is not suggestive of inability. – BrianH Aug 26 '19 at 22:58
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    I'm noticing your use of the word forgetting, as if you were supposed to remember the proof. You're not. I'm a tenured mathematics professor, with publications in analysis, and I don't remember that proof either. But I have enough experience working with compactness and the other concepts that I can recreate the proof, or some version of it, if you give me a few minutes. (Choose a sequence of points that approach the supremum, and then pass to a convergent subsequence...) And I bet you can do the same, if you just keep calm. – Nate Eldredge Aug 27 '19 at 7:12
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    Do you do some TAing / teaching? Often people say it's a waste of time, but I think it would help a lot with your problem. Some say that you don't ever really understand something until you have taught it. – Federico Poloni Aug 27 '19 at 11:57
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    Anecdotal evidence and some introspection indicates that the more one specializes in a math topic, the less one is able to do simple math. I have completely lost the ability to compute discounts at outlet stores, but can tell you how to compute the VC dimension of complex hypothesis classes. Such is life... – Spark Aug 27 '19 at 12:03

Sounds completely normal for PhD level work.

Many people doing PhD:s are like this. Some people who are envious or feel threatened of your high level abilities may try to shame you or make you unconfident with your lack of low level skills. There really is only one advice to give for this...

Trust in yourself and don't let it discourage you.

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    Thanks for your comment. – Michal Aug 28 '19 at 21:37
  • most if the time people suffering this will only overcome it once they start lecturing. many of those who don't won't overcome it. – Rudi_Birnbaum Aug 29 '19 at 14:37

Let me suggest a reason that you may be experiencing what you are, and it has nothing to do with any lack of ability. But it may, I think, involve intensity of focus. And "brain freeze".

When I'm working on something intensely, I tend to separate myself from all other thoughts and concerns. I dive very deeply into what I'm doing. I tend to do it even when writing here, actually. If someone asks me a question - any question - I freeze up for a bit in order to change context an reestablish a new one before I can answer. Maybe this is all it is with you, also.

If you are thinking deeply about some research issue, large or small, you focus very closely on the problem and various potential solutions, evaluating pros and cons, etc. If someone asks you, right then, to go get the mail, you won't, perhaps, even hear them.

But some people need a bit of time, when asked a question, to establish a context in which that question can be answered. If the question has any complexity, so might the context, and it may take a while to get there. You are "struck dumb" by the question and the impatience of the questioner just makes it worse.

So, here is a thought experiment. Take the same question that "stumped" you when asked and consider how your thought process (and potential success) might have been different if you posed the same question to yourself while sitting quietly with no specific other focus. I suspect that I would be able to work it out fairly quickly, and fill in the empty parts sufficiently well. Perhaps you would also. But the questioner wants an immediate, well formulated answer, just as if you had memorized it.

But memorization is a poor tool in mathematics. You can't (unless you have a very differently organized brain than others) simply memorize all of mathematics, theorems along with proofs. A much better tool for a mathematician is insight that lets you re-assemble the concepts and why they are valuable and true. If you are doing valuable research, I suspect that you have such insight.

But a bit of advice about oral exams. Don't worry too much about having answers instantly on the tip of your tongue. Don't worry too much about wanting to "work out an answer". Doing so aloud can be valuable as it is obvious to the questioner that you aren't stuck, just working. Don't worry too much about making an error, but then, admit that the train of thought isn't going to be productive and say why. This reveals to the questioner that you have a good grasp of fundamentals, and can think productively even if the actual answer eludes you.

I once, taking an oral in Algebraic Topology (not my strong subject), did exactly the above, working out the answer to a question and not getting there. After a bit, I said something like "I'm sorry but this line of thought isn't going to end properly. I can't see the answer, but I can at least explain why this isn't working...". Later I was told that it was a great response. Like a lot of things, it is more important to demonstrate competence than it is to answer every question on demand.

Insight, not memory.

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  • Thank you very much for your answer. I do believe what you said, but i wish my supervisor thought like you. – Michal Aug 28 '19 at 21:39

I agree with all the supportive comments here and that the relationship between the rapid recall and a research career is poor. Your supervisor should not have shamed you or put you down due to your poor memory recall. Your question of whether you can improve or rehabilitate your memory is a difficult one and I think an irrelevant one as others have argued here.

However, if you think there is a bigger and significant neurological issue that needs and you want further investigation, then definitely do so. Not because of your professor's inconsiderate remark of your poor memory recall, but because you have other concerns or patterns that have impaired your life and ability to enjoy life. Many people are being investigated and diagnosed with neurodiversity issues at an older age. So full neuropsychological testing that can place your memory issue as either isolated or part of a broader phenomenon is useful. Find a psychologist who is able to do formal cognitive testing, ideally a neuropsychologist.

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