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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:

  1. 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.

  2. 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.

  3. 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.

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    $\begingroup$ Why the downvote ? $\endgroup$
    – katana_0
    Jan 7 '20 at 16:10
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    $\begingroup$ I, too, have a terrible memory, but that is exactly why I love mathematics so much. You don't have to remember anything so long as you understand it, because then you will be able to find the solutions again without having to know them by heart. What seems to me to help a lot is to stay at it. Don't just study some field, make the exercises, and move on. Most importantly, consult your professors. Have you tried asking professors if you may attend their lectures? I think you will find they are a lot more open to having interested students there than you might think $\endgroup$ Jan 7 '20 at 16:21
  • $\begingroup$ @Interestedstudent I did; the profs are OK and even very welcoming with me sitting in their course but each of the courses are clashing with (atleast) one of the mandatory first year course(s) and my faculty advisor is not letting me (or any first year, for that matter) change the schedule/ditch first year classes to audit an optional course. $\endgroup$
    – katana_0
    Jan 7 '20 at 16:26
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    $\begingroup$ You could try to get some specific questions about the bigger picture in any of those fields you are studying yourself, and posing them to the relevant professors when and if they have time. They're grown ups, they'll tell you if they don't have the time, but seeing as they are generally passionate about their fields, I think they'll like helping you. $\endgroup$ Jan 7 '20 at 16:29
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    $\begingroup$ Now posted @ MESE. $\endgroup$ Jan 7 '20 at 20:24