Qfyilyi

It shames me to admit it, but I feel like I still haven’t figured out the right way to efficiently read papers and textbooks. I am a second year PhD student in pure mathematics, and I struggle a lot to balance the need to learn a lot new mathematics with the fact that I have a finite amount of time in which to do it.

I should have asked this question years ago: what is best (i.e. most efficient) way to read papers and textbooks in order to have a working knowledge of the subject?

Maybe I should be more precise.
I am working in both algebraic geometry and homotopy theory (think simplicial presheaves and K-Theory), and I have recently found myself overwhelmed by the amount that I need to learn. In an ideal world I would read background material by doing every exercise in every textbook, and by working carefully through every proof in every paper, but I worry that I just don’t have enough time. On the flip side, I often find myself “reading” mathematics without actually absorbing any working knowledge, so I basically don’t make any direct research progress.

I understand that reading in great volume is still constructive, and I have certainly learnt a lot about how mathematics fits together, but when I actually need to do new mathematics I consistently find myself lost.

To rephrase my question: can anyone offer an advice on their workflow when it comes to learning new mathematics?

Perhaps the way I feel is more or less how everyone feels, and it is confidence and organisation which is the problem. If that is the case, then I ask: can anyone offer advice on how to organise oneself day to day in a PhD to be productive? Moreover, can anyone offer advice on how to break out of a lack of self confidence when it comes to doing research mathematics?

I apologise in advance if this question has been asked many times before, but I haven’t been able to find the right thread. I apologise also if my questions are too multi-pronged, I just feel that they are all to interconnected to be split up.

I'm a second year student too.

I think that one have to study what really needs to know, for instance, if I need the Feit-Thompson Theorem (for some reason) I cite and use, if you need to use the tools and ideas that were used in that long proof, then you read the paper. Now, how to read it, in first instance, keep the central ideas and go deeper if you really need it for your work. There is a lot of mathematics, and nobody can learn everything, even if you only study algebraic geometry (it is a broad field). My tutor says that one will have a lot of years to study and go deep in topics that are of our interest, and in the moment, work hard fort getting the PhD (it is not easy, we know it), and publish

If you already have a good sense of how the relevant fields fit together, and you know your way around the standard references, I personally find that the best approach is to just jump into a good paper. Really studying a paper related to your research is a much better way to learn specific techniques than grinding through textbook exercises. I think there's generally too much emphasis on "background" at the expense of working on new mathematics.