I'm a recently admitted in a masters program in mathematics. My question is about "strategy" of learning. Should I focus on learning many things or should I pick an research area and start learning only stuff about that area? This is a problem because some scholars told me that if I want to get hired in academia, I must start publishing as early as possible so I better start to deepen my knowledge in certain area. However, actually I have not decided with area I fit the most, and I think that broadness is useful as well, because diversity is a necesity for creativity, just as professional musicians doesn't just hear say classical music, but rock music, or latin music to get inspiration, I think that mathematicians must not just read about their own area, but related areas as well.

In your experience, which "strategy of learning" is better for a begginer in graduate school? I must say that I have freedom for choose courses.

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    I think this depends a lot on your own personal goals and interests as well as your abilities. – Alexander Woo Jan 21 '18 at 22:35
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    What country is this? Graduate education differs substantially around the world. – Nate Eldredge Jan 21 '18 at 22:48
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    The next Terry Tao or the next me? If the next Terry Tao, how realistic is this? If it turns out you can't get a job that gives you enough time (whatever that means for you) for research, would you jump off a bridge, or do you have secondary goals you might want to consider? Do the areas you are interested in require knowledge of a wider or narrower range of mathematics? – Alexander Woo Jan 21 '18 at 22:53
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    Deep enough to prove something; broad enough to set the new results in proper context. – JeffE Jan 21 '18 at 22:57
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    @JeffE: That's an appropriate goal by the end of a PhD in math. For a masters (in a US-like system) I would set the sights lower: enough depth in core areas that a PhD admission committee is convinced you are adequately prepared. – Nate Eldredge Jan 21 '18 at 23:56

This isn't a question with a correct answer. I think most people who've been around math departments for a while have seen students go way too far to one side of this or another: students who think they know the first week of their Ph.D. the only field of math in which they want to work, and students who seem to constantly be learning new things, but never seem to settle down and concentrate on something. Lots of different strategies can work, and there's a lot of luck in how these things work out.

If you plan to do a Ph.D. in the US, then I would think about things this way: while you're doing your masters, you should be figuring out where you want to do your Ph.D. and preparing yourself for that. If you plan on going to another school, there's not too much point in specializing before you get there, since don't know what advisors will be available there. Taking some advanced courses, and choosing a brand area is fine, but you can feel comfortable trying to get a broad base.

You can reasonably expect to have 5 years to finish your Ph.D. after you arrive at the school where you will do so (so after you finish your master's degree, if you plan on moving). If this is the case, then, I would worry if:

  • at the end of your first year, you don't know what broad area of math you'd like to work in (number theory, PDE's, etc.).
  • at the end of your second year, you don't know who your advisor will be.
  • at the end of your third year, you don't know what the underlying objects in your thesis will be and how to work with them
  • at the end of your fourth year, you don't know the statement of the main theorem of your thesis

You shouldn't take this as gospel (at some schools expect students to finish in 6 year, so you can sneak an extra year in there somewhere), and of course, being worried doesn't mean that things can't be fixed, but what you can relax if you haven't done yet.

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  • I like this answer but I'd wish to see some tie-in with strategy for exam prep. Apparently OP's department is not very structured, so OP will have to do some self-structuring. – aparente001 Jan 22 '18 at 5:24

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