Question

Do people have the most difficult time learning their favorite field? How do you deal with that? It's happening to me in algebra and I'm sad. Is algebra just hard for everyone or it's just me? Did I pick the wrong field?

Answer 1

Perhaps you just care more about algebra and hence only are satisfied once you “fully realize” algebraic concepts, whereas when you consider other fields you don’t hold as high of a standard. This is what I found to be the case after some self reflection or, maybe, I just tell myself this.

Answer 2

Sometimes finding something hard just means you care enough to know why it’s hard

I wouldn’t worry about finding it challenging to learn your favorite field. I bet that if you pursued geometry or analysis to a similar depth, you wouldn’t be so quick to assume you were much better in those areas ;P

Just make sure you can allow yourself to feel stuck guilt-free, fighting through that extra stress is a recipe for burnout

Take a deep breath, you’re doing fine -    you’ll be surprised what knowledge you’ll amass over time.

Answer 3

It’s pretty common for people to either feel intuitively comfortable with Analysis or Algebra, but not both.   They’re very different fields with different basic approaches for introducing concepts and  tackling problems.  

I’m the opposite of you, algebra felt natural to me.  That giant pile of definitions and abstractions sat neatly in my head when I was taking it, I rarely had to look things up, everything felt graceful, well motivated and complete.  But analysis just never made much sense to me.  No matter how I looked at it, I was never sure *why* we were doing what we were doing at each step, why not something else, where it was leading, etc.  It all felt like arbitrary garden paths headed somewhere disquieting.  I got through the lowest-level analysis course that counted for my degree track, with a hefty dose of mindlessly aping the examples from the book.

I’m sure a cognitive psychologist could say something useful about learning/thinking styles here.  But practically speaking, I think you just sort of get through the one you dislike and then focus on the one that fits best.

Answer 4

Your reaction to algebra, compared to other fields, matches mine. So I went into differential geometry.

Why do you like algebra? In any case, just because you find it hard doesn’t mean you should give up. It just means it’s a deeper subject that requires more time and effort before you will be as good as you are in the other fields. Most people going into those subjects need more years of study than those in other fields. Harder doesn’t mean you can’t do it.

Answer 5

I believe that you can build intuition over time, with enough practice. I found differential geometry to be among the hardest topics I learned about, but I was curious and wanted to go into it anyway. After a master’s, it made a lot more sense. After enough time passed, I realized that I now find it easier than most other areas.

All that to say, don’t be discouraged. All math is very difficult at the research level, and I believe that it’s best to choose topics you’re genuinely curious about, so that spending many hours on them feels fun.

That’s just my 2 cents as a PhD student though, and ultimately you’ll need to decide for yourself if you want to switch to something else or not.

P.S. I dare say algebra at the level of algebraic geometry abstractions is hard for almost everyone when they first encounter it.

Answer 6

I think the responses here are all reasonable and don't have much to add there. I am curious for OP though - *why* do you like algebraic fields over other fields? I feel like the post doesn't offer much insight why, but that might be useful in diagnosing what's going on.

Personally, for me it differs field by field, though I definitely felt right at home with "and I never understand cohomology, they all seems like random calculations that works," heh. I actually feel like geometry or analysis come a bit more intuitively (and in undergrad, my true love was combinatorics) but are less satisfying to work in. Working in algebra on the other hand is less predictable and at times feels more like dealing with a logic puzzle than anything else, which I quite enjoy. Though it differs by subfield - I dislike working with groups in particular, but say for theory of k-algebras (in particular group algebras) I have a lot more working intuition.

Answer 7

In addition to what other people have said, it's also possible that you're taking objectively harder algebra courses. I was in my second semester of commutative algebra at the same time as I was taking point-set topology; one required page-long proofs that contained interesting ideas while the other had exercises that never went beyond a paragraph of unraveling definitions. Of course, I found one significantly harder than the other - it actually *was* harder than the other!

It's possible that you're comparing very advanced algebra to somewhat intermediate courses in other subjects.

Answer 8

I am only a first year grad student starting to do research in some part of arithmetic geometry, but here is my impression. Imo this can be an area that builds on many many theorems that are technically very difficult but there is usually a couple of guiding examples that motivate why this should be true, e.g. I can think about canonical models of Shimura varieties, and the modular curve case. And most people don't learn these technical proofs, unless you really have to (or you are someone like Brian Conrad), and that works fine for many people. About the example of cohomology, for me it helps having the guiding examples of geometry over R/C, e.g. all the comparison have an algebraic analogue. Also different people will have a different intuition for a certain cohomology theory, and my impression is that many of the new ideas come from testing things on the easiest examples available.

Answer 9

Why is it your favorite field if you so much stress around it?

Answer 10

Hah I felt the same way about algebra. That’s why I went into PDE and hard analysis.