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My apologies for such a general question, but as a 'math enthusiast', I've often wondered what the optimal strategy is to increase my mathematical maturity. Broadly, should I aim to hunker down & study one topic in detail (measure theory/analysis, which is what i'm doing right now), or should I study a broad swathe of different topics?

Also, are there specific fields of math that are better for developing mathematical maturity? I've often wondered whether number theory/graph theory/combinatorics are better for developing mathematical maturity as even basic proofs need one to think imaginatively.

My apologies again for such a broad question that will likely have many different contradicting responses. However, I don't have anyone to discuss this with, so thought I'd throw out my question here.

Tejus
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    Can you give us a better idea of where you are as a mathematician? We could probably give a better next step with this information. – ml0105 Apr 18 '14 at 06:09
  • Write proofs. Get feedback on them. – Potato Apr 18 '14 at 06:10
  • I know a good bit of point-set topology (Munkres), some algebra (Herstein). Really enjoyed Baby Rudin. Currently trying to learn measure theory. I've enjoyed combinatorics/graph theory/number theory but don't really know too much other than the basics there. – Tejus Apr 18 '14 at 07:00

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Hint: Try to teach what you learned (also what you want to learn) to people with different background!

Arash
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