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.