phd - How to be confident that my background for my research is adequate?

I am a third-year PhD student in Computational Algebra with a B.Sc. in Computer Science.

While doing my research, I often need to check my old textbooks for definitions, and I am concerned that I may not be adequately prepared despite having done the course work seriously. Due to this to me I feel like I should read a textbook for 2 weeks then start doing the research.

Others have suggested to only read a book when you need a particular lemma/proof/etc., however I am conflicted on whether this is good advice.

Question: What should I do?

Please note that my supervisor told me that you know enough for research. Is this kind of feeling common among PhD students?

Answer

If your primary symptom of concern is that you lack an encyclopaedic knowledge of your subject matter then I wouldn't worry - it would take a rare specialist with a very special brain to have that level of knowledge of an academic subject. Checking definitions is common in research and it is rare to memorise academic material in a level of detail that would render checks unnecessary. For most academics it is usual --over time-- to develop a solid knowledge of key concepts and results, with the ability to check more tangential matters, or particulars of definitions and proof, by reference to academic texts and papers.

The best thing you can do is to prioritise the importance of concepts and results in your field, and try to commit important core concepts and results to memory as well as possible, while making note of where you can get information you are likely to forget. Unless you are going to become a specialist mathematician it is generally unnecessary to memorise proofs of theorems, and even for people in this category, they might know a few proofs off by heart, but for most they will remember that a certain technique is used, without remembering all the details. For mathematical work it is useful to remember the substance of important theorems and remember the techniques that are applied to prove them. It is perfectly okay to forget the exact conditions of theorems or the exact details of proofs, but if they are really important, you should try to remember the substance of them, and have the capacity to find the details when needed.

Over time you will find that you learn your field more comprehensively and you become more used to the methods and techniques used in that field. For a mathematical subject you will usually find that most things are proved with variations of a relatively small kit of proof techniques, and you will start to recognise them well enough that you can remember roughly how to prove things without much cognitive load. As a result of this you will find that you can absorb material faster and so it will become much quicker to refresh forgotten knowledge with textbooks and papers. A refresher that took you two weeks at the end of your PhD might take only a single morning once you are an experienced researcher. That takes time!