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PhD students usually go through a specific topic in depth for several years: understand it and contribute to its literature. Some topics are multidiscipline by nature. Thus, while studying, students may find nice contribution opportunities in not-directly related areas to their research problem.

is it better to be goal-driven (i.e focus on the thesis problem alone) or opportunity-driven (i.e spend your PhD in different problems you encounter in this topic)?

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    You're proposing a false dichotomy. They should be both.
    – JeffE
    Dec 22 '13 at 1:55
  • @JeffE what if you simply cannot do both I mean deviating from your problem and investigating an opportunity takes several months from your PhD time
    – seteropere
    Dec 22 '13 at 5:40
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    Yes, indeed, time is limited. Make notes of incidentally-found opportunities for later, while most likely taking care of the main project. Dec 22 '13 at 14:53
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    Not all people or topics are the same. This depends on the person and the topic, on the opportunities, the goals, the personality, the skills, etc. The only correct answer is: "it depends, should be analysed on a one on one basis". You can make a decision tree with some inequations, margins, areas, etc. That's as far as you can get before generalizing too much and trying to apply something that works in some cases to cases where it doesn't work. Needles to say, the measure of "not-directly related areas" and "different problems" are subjective.
    – Trylks
    Dec 22 '13 at 15:53
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Although naturally the details of the landscape differ from subject to subject, here is my picture from mathematics:

In addition to the obvious point that one should "both" stay focused on a long-term project and be alert to incidental opportunities, I would claim that the best kind of thesis projects (and projects in general) are those that help create incidental opportunities along the way to a worthwhile large-scale goal.

(This in addition to "educational" benefits of well-conceived projects.)

The obvious hazard is to get distracted from a reliable, long-term project by too-speculative "cooler" possibilities one encounters along the way. The hazards here are like "... but don't quit your day job" as advice to aspiring rock stars, etc.

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Graduate students should be somewhat opportunistic. How far afield from their "home base" they should travel, however, is a different question that depends upon the expectations of their field, and what is allowed by the terms of their appointment.

For instance, a student who is entirely supported by a professor's research grant to study a particular topic will have a much harder time justifying spending lots of time on external problems that may be interesting but might not be covered by the grant in question. If such an arrangement is desired, then some sort of negotiation with the faculty member is probably needed to make sure everything is handled in above-board manner.

On the other hand, someone who is entirely supported by external fellowships that do not have funding "strings" attached is much more likely to be free to pursue whatever opportunities may present themselves.

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