TITLE: How to escape the inclination to be a universalist or: How to learn to stop worrying and do some research.
QUESTION [165 upvotes]: As an undergraduate we are trained as mathematicians to be universalists. We are expected to embrace a wide spectrum of mathematics. Both algebra and analysis are presented on equal footing with geometry/topology coming in later, but given its fair share(save the inherent bias of professors). Number theory, and more applied fields like numerical analysis are often given less emphasis, but it is highly recommended that we at least dabble in these areas.
As a graduate student, we begin by satisfying the breadth requirement, and thus increasing these universalist tendencies. We are expected to have a strong background in all of undergraduate mathematics, and be comfortable working in most areas at a elementary level. For economic reasons, if our inclinations are for the more pure side, we are recommended to familiarize ourselves with the applied fields, in case we fall short of landing an academic position.
However, after passing preliminary exams, this perspective changes. Very suddenly we are expected to focus on research, and abandon these preinclinations of learning first, then doing something. Professors espouse the idea that working graduate student should stop studying theories, stop working through textbooks, and get to work on research.
I am finding it difficult to eschew my habits of long self-study to gain familiarity with a subject before working. Even during my REU and as an undergrad, I was provided with more time and expectation to study the background.
I am a third year graduate student who has picked an area of study and has a general thesis problem. My advisor is a well known mathematician, and I am very interested in this research area. However, my background in some of the related material is weak. My normal mode of behavior, would be to pick up a few textbooks and fix my weak background. Furthermore, to take many more graduate courses on these subjects. However, both of my major professors have made it clear that this is the wrong approach. Their suggestion is to learn the relevant material as I go, and that learning everything I will need up front would be impossible. They suggest begin to work and when I need something, pick up a book and check that particular detail.
So in short my question is: 

How can I get over this desire to take
  lots of time and learn this material
  from the bottom-up approach, and
  instead attack from above, learning
  the essentials necessary to move more
  quickly to making original
  contributions? Additionally, for those of you advising students, do you recommend them the same as my advisor is recommending me?

A relevant MO post to cite is How much reading do you do before attacking a problem. I found relevant advice there also.
As a secondary question, in relation to the question of universalist. I find it difficult to restrain myself to working on one problem at a time. My interests are broad, and have difficulty telling people no. So when asked if I am interested in taking part in other projects, I almost always say yes. While enjoyable(and on at least one occasion quite fruitful), this is also not conducive to finishing a Ph.D.(even keeping in mind the advice of Noah Snyder to do one early side project). With E.A. Abbot's claim that Poincaré was the last universalist, with an attempt at modesty I wonder

How to get over this bred desire to work on everything of interest, and instead focus on one area?

I ask this question knowing full well that some mathematicians referred to as modern universalists visit this site. (I withhold names for fear of leaving some out.)
Also, I apologize for the anonymity.
Thank you for your time!
EDIT: CW since I cannot imagine there is one "right answer". At best there is one right answer for me, but even that is not clear.

REPLY [8 votes]: One thing that might help is to think about this advice as saying you need to develop new skills rather than being about the one true way to do mathematics.  That is, you have a lot of experience and skill at learning math systematically from the ground up like you did in your classes, but that's only one skill out of many that are necessary to be an effective researcher.  You also need to hone your problem solving skills, you need to learn how to identify appropriate interesting problems and research programs, you need to learn how to efficiently wade through the literature to find what you need, etc.  You have less practice at these things, so naturally you're not as good at them and so it's going to feel uncomfortable, but that's normal when you're learning new skills.
In other words, you're focusing too much on being well-rounded in terms of subject matter, but you're missing out on being well-rounded in terms of your skills.
That said, I do think it's worth following your interests, provided you still have enough time for your main research program.  There's more time in a day than my brain can handle working hard on my main research program, and the more of the rest of the time that was spent learning other mathematics, filling in gaps in my background, or working on a side project, rather than say playing 2048, the better.  So keep learning broadly and deeply, but don't let it eat into the time you're getting your main work done.