Feelings of inadequacy as a PhD mathematician [closed]

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!

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.

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!