Question

When is hard work simply not enough?

Answer 1

I think there's more to it than time and effort. Equally important is how you go about it. Office hours, collaborating, reading multiple texts, adequate preparation etc.

It is highly likely too early to be thinking about reaching a natural limit.

Answer 2

>  I never really internalized a lot of the concepts despite going over the definitions over and over again while attempting to write solutions.

Did you spend at least as much time going over *examples* of the new concepts you met?

Answer 3

First question: Did you come into this class with adequate background? For a first foray into graduate level courses, the stated prerequisites are enough at a content level, but the most important thing they require is the mathematical maturity of a graduate student. This means that you've seen "one undergraduate degree" worth of math across many different disciplines, and in turn, have seen a multitude of proof techniques. If someone's seen and done thousands of proofs that are loosely related to representation theory, then they'll probably do well in the class and see arguments faster and easier than those who don't.

​

To add to this, the first time people take a graduate level course they typically don't do all that well. I barely got through algebra when I took a graduate sequence as an undergrad, but a sailed through it as a graduate student. I didn't magically become smarter, but I just had 2 more years of mathematics under my belt and saw the concepts before (though I didn't internalize much the first time). You taking this course (and failing) *is* you putting in the time and effort to succeed, and I'm sure the next time you take this course you'll do a lot better.

​

Edit: I'll also add that, as a general rule, math is not a spectator sport. Understanding definitions is by far the easiest part. Doing computations (no matter how easy) is absolutely imperative to understanding the material. Next time you take a graduate level course, try doing the proof in a simpler case and see if it generalizes first (hint: it usually does) rather than tackling it outright. That's how most of the people solved these problems, I'm guessing.

Answer 4

You likely need a colleague or mentor to discuss these things with. Somebody reframing a concept colloquially for 2 minutes might help you more than 10 hours of looking at the definitions.

Change your idea of studying so that it doesnt involve only looking at a page for a long time and struggling on one given proof for a long time.

Come up with your own examples while out for a walk.

Answer 5

That point can come at any time. There are students who never get stuck, but they’re the unusual ones.

Did you try to work together with another student, preferably some who had similar struggles? Did you try getting help from the professor?

I suggest that when you take courses where there will be an official record of your grade, avoid courses where you suspect you might struggle like this. If you still want to learn the material, try to work unofficially with another student or two.

Answer 6

Representations are usually defined as group homomorphisms from a group to a space of linear transformations, but in fact they are perceived by mathematicians as modules over a group ring. If you don't internalize the concept of a module in this context, then indeed you will miss out on a lot of the "logical connections" and "mental flourish" that you are seeing from other students.

You can't get this sort of "clarity of thought" just from going back and forth through defintions and theorems. You need to do lots and lots of problems successfully (not just spend hours on problems without progress -- but actually gain the insights that the problems are intended to teach you). If you're stuck because you can't get through even a single problem, then you need some help getting unstuck. Talk to your professors and fellow students. When you see proofs written by your classmates, try to re-cast these proofs from both perspectives: representations as a group homomorphism, and representations as modules. Become fluent in both viewpoints so that you can translate back and forth between them without effort. Ask classmates and professors specifically to explain their thought process and how they arrive at the proofs in their minds, not just the written proof itself.

Representation theory is really hard. For some people it's the first hard subject in math that they encounter. (For other people, the first hard subject is differential geometry, or algebraic geometry, or functional analysis.) Learning hard topics in math requires more than just time and effort. You need time and effort, intelligently applied, with accompanying insights and intuition. Talking to people and asking the right questions is a far faster way to achieve insights and intuition than reading from a book, but asking the right questions is a skill unto itself. Even if you are talking to people and asking the right questions, it still takes time and effort, but not an absurdly superhuman amount. Also, if you've learned one of the hard subjects, then you kind of have an idea how to do it in general, which helps the second time around, and then the third time around, etc.

I don't think learning math is beyond you, but I do think you should reconsider your approach. Internalizing math comes from experience, practice, and usage, and can be attained more quickly by talking to people. Reading a book is something that you do only when there is no other choice.

Answer 7

Talent matters for research and don't listen to those who say otherwise, but you can base a career on a lot of hard work and not so much talent.