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I am a new graduate student majoring in pure mathematics. In past four years, I was interested in analysis, especially complex analysis. Now I think what attract me most are topics about algebra and geometry.

Lately I am reading some books such as “introduction to smooth manifold” by Loring W.Tu, GTM9 and “commutative algebra” by Aliyah. I read these books using the way of study analysis. I check every proof and every sentence, and write my notes.

In fact, I think I have a lot of harvest but the problem is: comparing to my peer, I think I known little. I haven’t studied category, algebra topology and differential geometry, which makes me feel nervous. At the same time, I found my peer’s study is very fast. For example, in the latest discussion, a classmate introduced the model category. I knew he just spent one day to read a book’s several chapters. But it looks like he was very confident in his knowledge about this theory.

By contrast, for example, I think I learnt about the core of tangent bundle, vector bundle and vector field using just one day, but I need three days to complete the all proofs and write my notes.

Should I switch my attitude and standard to learn more fast in order to get in touch with the advanced knowledge?

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    $\begingroup$ "rapidity doesn't have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant. Naturally, it's helpful to be quick, like it is to have a good memory. But it's neither necessary or sufficient for intellectual success." (Laurent Schwartz, A Mathematician Grappling with His Century, p. 31) $\endgroup$ Commented Oct 1, 2024 at 11:38
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    $\begingroup$ Related: Pacing for learning new material (mathoverflow) AND Do researchers actually read all of the papers they cite? (Academia SE) AND How long should it take to read a paper? (Academia SE) AND How to read an article and make it actually useful? (mathoverflow). $\endgroup$ Commented Oct 1, 2024 at 13:04
  • $\begingroup$ What would be the result if you did read roughly, without trying to understand every detail? Make a first reading as rough as you like but who could doubt the need to go back as many times as it took to understand every sentence? $\endgroup$ Commented Oct 18, 2024 at 20:56

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General comments:

  1. Knowing the basics of your subfield really well is the most important thing for your future success. If you have that foundation, you can build whatever you want on top of it. Grad school is the time to dig deeply into your field--you'll have much less time for this later on.

  2. Having said that, it's good to supplement these deep dives with some more surface-level studying (like Wikipedia-ing definitions and going to the math library and reading introductions to various graduate textbooks) so that you have a sense of how concepts fit together broadly. It will also help you to communicate with other people with different specialties. You don't need to spend a huge amount of time on this, however. Point one is ultimately more important.

  3. You have to find a way, psychologically, to not compare yourself to other people. (Even as you learn their good habits by example.) A competitive field like math academia will drive you insane otherwise.

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