I am a first year undergraduate student, currently in second semester.
So basically I learnt most of the first year stuff in high school, so I have a lot of free time in this year (currently in second semester), and thus I'm self studying math in the free time. In the first sem I tried reading some measure theory and lebesgue integration, some galois theory and some algebraic topology (I think I had the prequisites -- I did half of baby Rudin, most of Artin and Topology, Willard in high school).
So here are my problems:
I tend to pick up very little "big picture" intuition while self studying. I was learning measure theory from Stein Shakarchi, and my experience - it's hard to describe in words - was like I can follow the theorems as a standalone basis, I don't have much difficulty solving the exercises but I was heavily bogged down in the details so I could neither see the big picture nor any non-trivial/deep connection between what's done in a page and what was done say three pages ago.
I have an extremely terrible memory. My memory is so bad is that while I recall working on an abstract algebra book till Fundamental theorem of Galois Theory around in October, now I forgot what's a simple field extension (or even worse, I forgot the exact statement of Fundamental Theorem of Galois theory ! Only thing I remember is that under some conditions on $K \subset L$ it gave a one to one correpsondence between intermediate fields between $K$ and $L$ and the subgroups of $Gal(L/K)$, and one direction is not hard but the other requires linear independence of characters)! This is extremely frustrating, when you forget what you learn one/two months ago completely.
I was a relatively good problem solver in high school (did quite well in national/regional math olympiads) but my problems solving skills don't seem to be that good in college level math. For example, I tried proving Hilbert's Weak Nullstellensatz (when I was studying it from Artin) couldn't prove it even though the proof is "short". If you say it's short but tricky -- well I couldn't even prove the fact that the characters are linearly independent (I recall trying for like 30 minutes and then being frustrated since it seemed a pretty easy proof and then seeing the proof). Due to #2 and lack of mathematical maturity, I tend to give a shot at the proof of some theorem myself before seeing the solution, but this way consumes a huge amount of time and also I fail to prove the theorems when it's nontrivial in most of the cases.
So basically I still have plethora of time in this semester (just started; four months long) and I plan to self study some complex analysis, manifolds and algebraic number theory but the points above are heavily discouraging me to self study math or pursue math in general. Whenever I start to read a book I feel depressed as the pointlessness of me studying a book since (a) I will not have any big-picture understanding of it and (b) I will forget what I did after a while anyway (if I don't use it).
Also I am currently the topper of my batch (which consists of one IMO medalist and many national olympians) -- but that's solely because I knew most of the course materials beforehand so I didn't need to put in too much effort outside the class to understand the stuff. As I mentioned in the points above, it's hard for me to get used to and not forget new and hard math so sometimes I feel insecure about performing extremely terribly in the upper division courses which I didn't know a priori (basically I wouldn't have any edge in those courses), so this is discouraging me more to study maths.
Am I making any obvious mistake while self studying ? Is there any global change which I should make to end up with much better understanding/retention of the material ? Would be a wise choice for me to not stay in academia and move on to CS/other applied math courses ?
Note: People can suggest actually sitting in the lectures of the courses I want to self study, but that's not feasible for me: While most profs are friendly towards second year people or higher auditing any course (basically sitting in the lecture w/o crediting it), they're mostly cold towards first year people sitting in the lectures.
[Crossposted from Math SE]