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As an undergraduate engaged in applied mathematics research, specifically within mathematical programming, my work has quite some intersection with areas like algebraic geometry for polynomial optimization and the use of manifolds in manifold optimization algorithms.

Producing research is my priority, but I'm afraid learning mathematics by skimming through many basic concepts will undermine my foundation. For instance, a theorem statement such as Hilbert Nullstellensatz will be useful, but its proof is not trivial, and looking deeply into every theorem I encounter is simply infeasible, and usually not helpful for research.

I'm contemplating whether to quickly get a grasp on the main theorems (by skimming), make a list on what I don't understand, and revisit unclear concepts later. However, I worry that repeated superficial reviews might lead to a false sense of understanding. Should I be concerned about this, or is there anything I can do?

I read about this post some time ago. I agree with not delving into the rabbit hole, but what troubles me is that I don't want to do mathematics in my whole life as if it were magic. Even without knowing the proof, I think I should at least have an intuition on how the proof works, and if necessary modify the proof so that it suits my need for other things. However, I think I often would accept myself just applying magic, it works so well in some cases that I don't even bother to look further. But applying magic is never a natural process, I have to think of "applying" the theorem rather than having the theorem itself appear naturally, (kind of like plug and chuck formulas to see which one works). Therefore, I really want to be conscious about what I know vs what I've just memorized.

wsz_fantasy
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  • Are you working with somebody more experienced? This is exactly the kind of question you'd want to ask a mentor. The level of understanding needed can vary greatly and somebody familiar with your field would have an easier time advising you. Of course it would be awesome if would know everything, but the day has only a finite number of hours. – Severin Schraven Feb 19 '24 at 21:10
  • @SeverinSchraven I'm working with a professor and one of his grad students. I guess I can ask them about differential geometry, but I think they are not too familiar with algebraic geometry. Plus, I don't know what they can tell me, if I memorize all the theorems, it might be the best approach for a very short amount of time but is not a good thing to do in the long term. Though I guess I don't know, I'll ask them next time we meet. – wsz_fantasy Feb 19 '24 at 21:19
  • The point is not that you should take everything as a blackbox, but for certain things it makes sense to do so. For example, I have been supervising a bunch of undergraduates on a research project where we have used some facts about Haar measures. Those were things they could just take for granted and it would not impair their understanding. However, they needed a better understanding of the more algebraic side (global fields etc). We also used a bit of algebraic geometry, but that would still not necessitate them looking at the proof of the Riemann-Roch theorem. – Severin Schraven Feb 19 '24 at 21:40
  • Similarly, if I was to supervise undergrads on spectral theory, I would not ask my students going through the proof of the spectral theorem. I'd take a minute to tell them the intuition, but it is just a technical tour de force and not really helping them. However, it is hard to gauge which parts you can omit, this is why I'd suggest asking more experienced people. – Severin Schraven Feb 19 '24 at 21:44
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    @SeverinSchraven I see what you mean, I asked them a little and learned that I don't need to know more about differential geometry besides about embedded manifolds, I will ask more specifically. However, my question is a little different, after my undergrad years, I might be working on a different project. I wonder how can I be conscious enough to know what I mastered vs what I did not. Sometimes there are tricks hidden in the proof itself, and it seems to me that at that time I might tell myself that "I already know this, so there is no need to read or learn more about this". – wsz_fantasy Feb 19 '24 at 23:24
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    I'm very afraid of leaving holes in my study that I'm not aware of. For the parts of mathematics I feel I'm better at, I often don't need to think about theorems statements as to purposefully apply them, the theorem themselves naturally enter my brain in my thought process, or I might even derive them on the way without realizing it is a theorem. So even if my professor tells me perfectly what to learn, I still have this concern of leaving unawarable gaps in my study. – wsz_fantasy Feb 19 '24 at 23:30
  • Well, you do have gaps (we all do). I would argue though that we are not "unaware" of those gaps (you and I do know that we don't know those proofs). Once you realize that you really need to understand the proof of something you can still sit down and work through it. The problem is more that you need to get a good intuition about certain things (which not necessarily come from the proofs). To stick with the spectral theorem above, you would need to understand that spectral calculus boils down to apply the function to the eigenvalues after diagonalizing your operator. – Severin Schraven Feb 19 '24 at 23:55
  • However, probably somebody more successful than myself should take a stab at answering your question :) – Severin Schraven Feb 19 '24 at 23:56

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