I'm interested in knowing how you manage your time when studying fields that provide useful background for your research, but aren't its primary focus. I feel a 'full' understanding isn't strictly necessary if you only need enough context to follow what's going on when elements of the field crop up in your research - at least, I hope so, since academia is vast and time is limited! Still, I suspect that simply attending lectures (or equivalent time spent reading) wouldn't bring sufficient understanding, as you would normally build on the lectures with revision/further reading and be actively solving example problems, in order to be able to understand later content.

As a baseline, let's say you want an understanding of a module for which 150 hours of total study are suggested and whose core content is covered in 30 hours of lectures, sufficient to:
- score 50% in every question on its written exam,
- follow in real time an explanation of the perfect exam solutions,
- spot when a technique/result is used in a research paper and understand well enough to continue reading after a few minutes of thought/referring back to learning materials.

With access to any lecture notes, problem sheets and textbooks you need for self-study, how would you go about gaining this level of understanding, and how long would you expect it to take in total? Does the answer change if you're returning to a module you already studied, but so long ago that you would fail the exam if you took it today?

Thanks in advance!

[Probably-superfluous context: I'm looking to return to academia with a PhD in Mathematics (my undergraduate degree), having done some postgraduate masters in Computer Science and Machine Learning and a few years in industry in Data Science. I'm interested in topics I didn't get to study as an undergraduate (e.g. Formal Logic, Set Theory, Model Theory, Modular Forms) and feel I need to study in some depth before committing to PhDs in that area. On top of these, there's a vast list of relevant modules that I have studied before, but which I feel so rusty in that I would almost need to learn them from scratch - but perhaps it will all come flooding back to me in a fraction of the original time. I'm trying to squeeze all this into the limited amount of time I have to study alongside working full-time.]

I am passionate to answer the question because I feel like my experience is relevant [I have a PhD in math but didn't study it in undergrad, so I had to do a lot of study during my PhD to catch up].

Before we get to "how to go about", I believe what is more important is that you choose the right modules to study in your limit time:
Be absolutely clear what you want to gain from studying each module. Is it for you to decide which direction to move into? Is it to satisfy the prerequisite of a particular PhD programme? Is it to understand a potential supervisor's area and write your proposal?

As for "how to go about":
You said you wanted to look at topics that you didn't study in undergrad, which makes me think you are not sure what narrow field you want to study and are yet looking at potential supervisors. In this case, I would say don't worry about research, just study the modules (the ones you've never studied but may potentially be necessary if you do a PhD) as if you are in undergrad. Personally, I would read the lecture notes (try to understand the logic between different statements, not just remember knowledge points) and go through the exercise sheets. In your case, it may even be beneficial to take online courses so that you can provide relevant experience in your application. You should find yourself comfortable with the technique and the way of thinking before deciding on a PhD in that topic.

If you have some ideas of what PhD you are interested in (even if not certain), you should start looking at potential supervisors and even reach out to them. A lot of supervisors do list prerequisite and references on their websites. It is always more efficient to follow supervisors' lead than jumping into the abyss yourself. When you try to digest these materials, the topics you need to study will surface, and then you can find the suitable lecture notes to help you study. [The thing is, different lecture notes and textbooks may go about the same topic very differently. You can be studying a module thoroughly but completely missing the bit of technique you need because that module is designed to focus elsewhere.]

Continued...

How long does it take? If it is an undergraduate lecture note, not long at all. I recall when I need to teach tutorials of a new module it probably took me at most 20-30 hours to prepare as the term goes on. For my own research, I rarely studied the whole modules, only the part I needed. I never studied it to the extent of scoring well in exams, because that would require practice and memorising, none of which was efficient for me as I was in my PhD already. If it is more advanced materials suggested by potential supervisors, then really as long as you need.

I would say, don't worry about reviewing modules you have studied but feel rusty in. If anything comes up in your new modules, go back to review that specific part. Nobody has that much memory to spare. I can't remember my own proof from two years ago. It is more important for you to know where to find a specific knowledge and how it fits in the bigger picture, then to be good at exercises and exams [A lot of profs are not so good at the exercises and exams of the undergrad modules they teach either.]

I am not familiar the specific topics you are interested in, but if you want (and depending on which country you are doing your PhD in) I can share a bit more of my experience.