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Short version: Graduate prelims include lots of topics which are only distantly related to your particular research area. How does one motivate oneself to learn and do this stuff?

Long version with personal background: I'm a 3rd year undergrad who's planning on applying to graduate schools in the US next year (in mathematics). I'm taking a graduate course (Galois theory) which, in the graduate programs I'm looking at, typically amounts to less than 1/3 of the material on the algebra prelim, which is one of the usually 3 topical prelims. For me, the concrete consequences for failing this class are academic probation with possible loss of funding/scholarships; if I were in the graduate school here, this would be possibly ejection from the program.

Unfortunately, I find this topic to be horribly uninteresting. I did very well in the first half of the class because it was essentially a very fast-paced redux of group and field theory, but I can't find a single drop of inspiration or interest in my mind for these classic Galois theory results that we're developing. I've done some undergraduate research in algebra (factorization theory) and, if I were to go into that as a professional, I feel confident that I will never touch Galois theory ever again. So for me, Galois theory being mandated to be learned by me for only this one purpose, a prelim exam. Therefore, I'm having trouble finding motivation to learn it.

The lectures are in the usual "lemma: proof, theorem: proof, corollary: proof" style (occasionally throwing in a sentence or two of motivation, why we study this topic), with weekly homeworks which heavily supplement the lectures (only a small percentage of grade), and 2 tests (midterm and fnial). These homeworks are pretty in-depth and are expected to take upwards of 5 hours to properly complete. (Usually 1-2 pages of proofs for each problem, and 3-5 problems per week.) Nobody is holding my hand any more and walking me through the class like in undergrad classes. (E.g., the homework problems aren't even close to self-explanatory and often cover topics not even mentioned in lecture.) Even though I'm showing up to lecture and taking detailed notes, I haven't even started the last 3 homeworks. The final is still 3 weeks away so it's not impossible for me to catch up, but it's going to take a lot of hours to get this material learned so that I don't fail the class.

As far as I can tell, this class structure is more or less standard in graduate mathematics, so if I'm serious about wanting to continue in academia I'm going to need to find out how to manage this problem (lack of interest). How do others handle this? Where does the motivation come from? (other than Adderall.)

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In this case, the best way to motivate yourself to study this uninteresting material is to try to become interested in it. It's important not to specialize too early, and as a third-year undergrad, you should only now be in the process of deciding a broad subfield of mathematics to specialize in (usually Algebra, Analysis, Geometry/Topology, Applied Math, or some combination of the above). Given that you're interested in becoming an algebraist, it's simply too early in the game to be crossing specific sub-fields of algebra off your list. Especially not Galois Theory, the basics of which you will very likely be expected to know if you take graduate courses in algebraic geometry, number theory, or the like. And even if you don't directly use this material in the future, it's a chance to develop mathematical maturity and intuition. (I'm glad to have been exposed to Galois Theory as an undergraduate, and I'm an analyst!)

As for this particular class, it sounds like it's simply going too fast for you. Which there's no shame in; I certainly had the same experience when taking graduate classes as an undergrad. If you revisit this material in a few years, you may be surprised to find yourself enjoying it. In which case you'll be thankful for anything you learned the first time around, even if it didn't fully sink in.

Basically, you should motivate yourself by keeping in mind that the effort you put into this class will benefit you in ways that aren't apparent right now.

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Where does the motivation come from?

Ideally, the professor teaching the class should do a good job of motivating the material, but it sounds like they somehow aren't connecting with you. For a topic like Galois theory, which is a standard part of the core mathematics curriculum throughout the world, you can be sure many mathematicians find it fascinating and important. If your class isn't successfully conveying the fascination and importance, look elsewhere for it (this class shouldn't define or limit your explorations). Talk about Galois theory with your friends and classmates. If it's clicking for them, they can help you appreciate it; if it isn't, you can investigate further together. Ask the professor questions. Talk with other mentors. Look for other books that you might find more engaging. Search and ask questions online. All this is much more work than just going to lectures and working on problem sets, but it's the only way to get a deeper understanding. (And five hours of homework a week is not much for a graduate course. This leaves you some time to study the material on your own.)

I've done some undergraduate research in algebra (factorization theory) and, if I were to go into that as a professional, I feel confident that I will never touch Galois theory ever again.

You may be right that you won't ever need it in your research. However, if you become an algebraist, there's a good chance you'll have to teach Galois theory someday. More generally, as a professional mathematician you'll be responsible for having a much broader knowledge of mathematics than just your own research interests.

Even though I'm showing up to lecture and taking detailed notes, I haven't even started the last 3 homeworks.

It sounds like a vicious cycle: your dislike of the material keeps you from working on it, but that limits your understanding and makes you dislike it all the more. The good news is that I'd bet you don't have an intrinsic dislike of Galois theory, but the bad news is that you'll have to break the cycle somehow. This is largely a matter of psychology. When I've been in situations like this, I've gotten out through indignation, along the lines of "I'm not going to let some crummy lecturer keep me from understanding and appreciating one of the highlights of mathematics." But what works best for you will depend on your personality.

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Good scientists need to have a fairly broad background in their area, the requirements of a department generally reflect what that department has decided needs to be part of that background in order for its graduates to be considered well educated in their area.

Learning a broader base of material is important for a couple of reasons. First of all, it gives you more intellectual "tools" for approaching problems in your area. Even if a particular subject (e.g., Galois theory) turns out not to be very useful for the problems you are working on right this very moment, it is likely to be useful and important for a lot of things in the discipline, and if you have some understanding of it, you will be able to recognize when you are dealing with an unexpected case where it is the right tool for the job. Much time and energy is wasted when people who are missing a piece of background try to re-invent the wheel. Likewise, you should understand it well enough to recognize when it is the wrong tool for the job.

Second, in most graduate classes there are two simultaneous lessons being taught: the material itself, and the structure of the science supporting the material. Galois theory is partly so hard because it is an example of a mathematically deep concept, and understanding how Galois theory works may also better enable you to recognize and develop another new deep concept.

Finally, science will always be throwing new ideas and concepts at you, and is also increasingly interconnected and interdisciplinary. You need to develop the skills to be able to learn and cope with new material, even if it is not a type that you prefer, or else you will quickly become locked into an ever-shrinking subfield and likely obsolete.

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Lots of good advice here. But I'm going to take a different approach.

You are not really interested in this material at this stage of your life. You might be later, but for now, you really aren't. It is unfortunate that your program doesn't allow you more freedom of choice.

So now it's time for Operation Get By. There are several things that can help you accomplish this:

  • Find a human being to help you. It might be the professor, it might be the TA, it might be a professional tutor, it might be a friend, it might be a really nice grad student.

  • Find a study group (it might be a bit too late in the semester for this -- but there's no harm in trying).

  • Try some internet research, to see if there are some solved problems out there that are close enough to your homework problems, that you can use them as starting points.

I was forced to take Assembly Language once, and I HATED it. I have never used what I "learned" in that class. I cussed and whined my way through all the projects. I hated the teacher. You don't have to love everything in math, or have a good fit with every teacher. But you do have to pass your required courses. Sigh.

What I am proposing you do is akin to faking it in music. Once I had to learn Tchaikovsky's Fourth in one week. There were masses of fast passages with lots of accidentals. The first couple of days, I did my best to plow through as much of the material as I could each day, trying to find sensible fingerings, practicing in a thorough way, to get the hand shapes comfortable. But by the middle of the week it had become clear that the conductor was NOT going to help. He was going to take the fast passages so fast they were UNPLAYABLE. He was not going to rehearse the most difficult fast passages carefully, to give us a chance to work up gradually to a fast tempo. So I gave up. I resigned myself to faking huge swaths of black notes. I have never done this before or since. But really, there was no other way out. I could have given myself lots of soreness in various body parts, practicing those passages like crazy, but the end result would have been no different -- so I cut my losses.

if I'm serious about wanting to continue in academia I'm going to need to find out how to manage this problem (lack of interest). How do others handle this?

Don't worry about this. In graduate school, you get to follow your own interests more. You'll have a wonderful advisor, who will make sure your own interests are not leading you down a blind alley.

Also, you will be a little older, more knowledgeable and more mature than you are now. Just take things one step at a time.

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