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tl;dr I would like to practice problems to develop a better understanding of topics in mathematics graduate-classes, but I don’t get time off my homework (which only covers a few problems), so I need a plan of study.

As the title suggests, I am doing math and there’s too much I want to do but not enough time. I am currently a sophomore (undergraduate), but I've been taking math graduate courses (namely algebraic topology, algebra II and an independent study in algebraic geometry) for the past two semesters. Apart from this I also have a computer science workload which isn’t too heavy as of now.

While I have been able to understand a lot of concepts really fast and get a working knowledge, I feel certain topics go off pretty quickly in classes. I believe the the true way to internalize all the its and bits and be able to work with them is by doing problems; lots and lots of problems. For instance I am not very comfortable with modules (which we covered in Algebra I last semester), and so am trying to do questions from Atiyah’s commutative algebra book (also useful for my algebraic geometry) nowadays alongside my courses. However each course has homework which takes a lot of time (especially cause it’s based on stuff you just learnt) and so it’s hard to make time.

I am really enjoying all my courses, but I want to focus more on problems (such as from Atiyah) to truly get better at working with these concepts rather than just knowing them abstractly. Staying up late at night works with me, but I have early morning classes so this is often not durable. I would really love some advice on how to plan because I don’t want to do too much math without being proficient in topics I do, and believe am a foundational stage so want to be careful.

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    You might want to add a tl;dr
    – whoisit
    Feb 27, 2023 at 10:06
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    Please prioritise your sleep, it does wonders. You're on a high speed train to burning out. It would also help to explain why you want to do "lots and lots of problems" in all subjects you study? I can understand why you'd want to do this for one or even two subjects, but not for every subject you take. Feb 27, 2023 at 10:06
  • @whoisit Done thanks Feb 27, 2023 at 14:23
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    @user3508551 I mean I want to get good at everything properly without rushing through theory Feb 27, 2023 at 14:24
  • That's the general problem of higher education: we don't give people enough time to practice and rush them through the program. I've been thinking about this problem a lot. The only solution I have found is to drop some courses. Taking 16 credits a term you sacrifice the quality of your education because there are only 24 hours a day which is not enough to practice. I would say 2 courses (8 credits) should be the maximum. You end up paying more, but you get high quality knowledge. Also, check out this post of mine to see how to plan the day: academia.stackexchange.com/a/188988/109799 Feb 27, 2023 at 16:30

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My advice is to change the frame of your problem. Right now, you are wondering how to cram more mathematics into the same amount of time. Instead, I suggest thinking about how you can make the best use of the time available. There are only so many hours per day, and far fewer of them that we can reasonably dedicate to thinking really hard.

First of all, supplementing your coursework with additional study is a perfectly fine thing to do. Working out problems is a great way to help learn the concepts, and using additional texts can help to clarify issues compared to looking at only one textbook. Your experience here can also help to direct your further study, when it comes to selecting topics within your degree work, and making career choices in general: paying attention to what you find interesting or boring, easy or hard, will yield good information. But burning yourself out by trying to solve all the problems no matter the cost, is obviously very bad.

Overall, "there is no royal road to commutative algebra", and we shouldn't expect it to be easy to learn. I'd like to propose to you that since the road is indefinitely long, and there are other roads to travel as well, it would be useful to do a few things to shape the study -

  1. Set concrete milestones for what you would like to learn. There is no limit to the time that you could spend, especially if you are just working through a textbook linearly from front to back, and judging for yourself whether you have a sense of feeling proficient at what you are doing. Instead, you should decide on specific topics that you feel are necessary, and on success criteria that are more measurable. Rather than "achieve total godlike mastery of algebra", the goals can be more like "be able to reproduce the definition of the field of fractions of an integral domain and prove its universality property". This allows you to not focus on other things, and also gives you a stopping condition for the thing you are focusing on currently (i.e. no matter how fascinated you are by the concept of torsion submodules, you must allow yourself to stop thinking about them at some point, or you will never do anything else). Feel free to discard topics ruthlessly; you can always come back to them if need be.
  2. Take note of your short-term progress. In pursuit of one of these milestones, at any given moment you might be stuck or making only very slow progress. It is surprisingly hard to notice when you are stuck. Very often, the correct tactic at these moments is not to carry on. If you are tired, hungry, etc., then you will be better off addressing that than desperately ploughing on with the mathematics. After a break you can also determine whether it is actually worth sticking with the specific problem. Maybe the right move is to work on something else, and return to the original issue later or never. You have still learned something, even if it's only that the topic is harder than you estimated. Or, now feeling fresher, you may come up with the breakthrough technique after all.
  3. Set a strict time-box. It's tempting to feel that the key insight is just around the corner. If you are having trouble with actually disengaging, then pre-commit to setting a timer (literally!) for when you are going to leave your desk. When the bell rings then it's time to drop the pencil. At times you may feel like you're racing against the clock - but another session will come along soon enough. The point is that by budgeting your time, you are not only protecting the rest of your life against mathematical intrusion, but also being more effective at using the resources you do have - diminishing returns set in pretty quickly, when it comes to slogging away.

What this boils down to is a change in mindset. Instead of trying to cram more and more study into a finite time, you treat your time as fixed and then carefully select what to do with it. Recall that even top athletes do not train all through the day and night. They make the best use of the time available and try not to destroy themselves in the process.

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I don't think you need to do more problems. You need to figure out how to work smarter rather than harder.

Grothendieck, generally considered the greatest mathematician of the 2nd half of the 20th century, did not know (or at least it's a believable story that he did not know) that 57 was not a prime number! He had tremendous gaps in his practical knowledge.

It's impossible to know everything; there is too much. You can't internalize all the its and bits of everything. What you have to learn is how to work with concepts even when you're not comfortable with them. (There are sections of some of my research papers where I understood the ideas just well enough to write those sections and have promptly forgotten how they actually work.)

Also, modern pure mathematics is built on layers and layers of abstraction. Certainly, doing a number of similar problems can help you grasp a concept that abstracts what is similar among all these problems. It's a good way to understand a single layer of abstraction very well. However, with multiple layers of abstraction, the difficulty of this approach of understanding by doing problems is exponential. If you need 10 problems to understand an abstraction, then you need 10000 problems to understand an abstraction of an abstraction of an abstraction of an abstraction. (And 4 is actually a small number of layers of abstraction for concepts in modern pure mathematics.) You're not going to do that. You just need to get better at understanding abstraction and working with it so that you don't need to work 10 problems - ideally you can understand an abstract idea from working only a single problem so that there is no exponential growth at all.

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  • 57 was not a prime number --- I don't have the primes memorized this high (but then I'm not remotely "a Grothendieck" either), so I'm guessing the point here is that he didn't know the divisibility by 3 rule, the first thing I would expect someone to apply when considering the primality of an odd number such as this (and which can be done in an instant to show divisibility by 3)? Feb 28, 2023 at 9:10
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    I think less memorizing the list, than having internalized enough familiarity with numbers that you "instantly" notice that 57 is 3 less than 60, and 60 is obviously divisible by 3, and so 57 must be as well - or enough relationships like that are at your fingertips that you'd be unlikely to regard 57 as prime, especially once some challenges you. A bit like a biologist who's asked to pick a reptile and says "ok, how about a squirrel" - they must be doing a kind of biology where the characteristics of real living animals are not at the forefront.
    – alexg
    Feb 28, 2023 at 9:59
  • @alexg: notice that 57 is 3 less than 60 --- In fact, I happened to think of this immediately when reflecting back on this while I was at the gym (between my last comment and this comment), and I regretted overlooking this explanation when writing my earlier comment! And your observations about "having enough relationships ... at your fingertips" and "not at the forefront" seem spot-on for the root issue here. Feb 28, 2023 at 11:44

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