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tl;dr: What advanced math subjects should I study to be able to learn new material more efficiently?

I recently finished undergrad as a computer science major, but in my current work there's a ton of content I need to learn (signal processing, advanced statistics, etc). I've always believed that one of the most important skills is the ability to learn new ones, so I'm interested in ways to learn more efficiently. My coworker, who is a math major, is also learning similar topics, but I noticed he understands everything he reads instantly. After doing some reading (like these similar posts), I figure it's because he understands how to think abstractly and approach new concepts not at face value but in relation to deeper core principles that drive a broad range of ideas he's already familiar with. In other words, I believe he can easily digest two seemingly unrelated concepts he's never heard of before by realizing the general, high-level properties they have in common with other concepts he's come across.

I understand that doing new problems/proofs, reading papers, and increased exposure in general are great ways of improving mathematical fluency. A large part of it is getting used to tough math and acquiring the prerequisite knowledge for more advanced fields, but I think the most important part of it is the ability to see the real meaning behind a problem. I feel like I'm still lacking...something fundamental, but I'm not sure what. I think it's that ability to passively model problems not just abstractly, but innately. For example, a "low" skill approach to a problem is to brute force or follow the obvious; a "medium" skill approach uses tools you've learned (e.g. applying tricks you've seen before in proofs); a "high" skill involves some intuition (e.g. intuiting a recursive substructure in a problem to use dynamic programming); and an "advanced" level is something I'm only vaguely aware of existing.

To that end, I'm interested in studying some core subjects to improve my mathematical thinking, problem solving, and ability to understand new ideas better. What do you suggest are good subjects to learn (and textbooks to read), either at the start or ones I should build up to? At the moment, I'm beginning with real analysis and complex analysis and trying to do the exercises as well. I'm also interested in learning abstract algebra, and eventually category theory (I had taken the course before but struggled a lot, so I need to strengthen the basics first).

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  • $\begingroup$ Intuition is built by training training training. Do math problems, read solutions by others. Throw as much stuff at your brain which your patience and pain threshold allows for. Eventually your neural network will start to see patterns. $\endgroup$ – mathreadler Sep 10 '18 at 18:55
  • $\begingroup$ Thanks for the response, but practice what in particular? "Math problems" is pretty vague, and clearly not all subjects are suitable. For instance, vector calculus is wildly different from category theory, yet the latter has concepts that help understand a wide variety of other topics, from abstract algebra to even object oriented programming. Whereas vector calculus is infinitely more useful in application, category theory is arguably better for the purpose of self-learning. $\endgroup$ – Booley Sep 10 '18 at 19:25
  • $\begingroup$ Don't be picky. Practice all kinds of problem solving you can get your hands on. You will find out what you have an aptitude for along the way. $\endgroup$ – mathreadler Sep 10 '18 at 19:29

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