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.
Best Answer
I think that, for the majority of students, your advisor's advice is correct. You need to focus on a particular problem, otherwise you won't solve it, and you can't expect to learn everything from text-books in advance, since trying to do so will lead you to being bogged down in books forever.
I think that Paul Siegel's suggestion is sensible. If you enjoy reading about different parts of math, then build in some time to your schedule for doing this. Especially if you feel that your work on your thesis problem is going nowhere, it can be good to take a break, and putting your problem aside to do some general reading is one way of doing that.
But one thing to bear in mind is that (despite the way it may appear) most problems are not solved by having mastery of a big machine that is then applied to the problem at hand. Rather, they typically reduce to concrete questions in linear algebra, calculus, or combinatorics. One part of the difficulty in solving a problem is finding this kind of reduction (this is where machines can sometimes be useful), so that the problem turns into something you can really solve. This usually takes time, not time reading texts, but time bashing your head against the question. One reason I mention this is that you probably have more knowledge of the math you will need to solve your question than you think; the difficulty is figuring out how to apply that knowledge, which is something that comes with practice. (Ben Webster's advice along these lines is very good.)
One other thing: reading papers in the same field as your problem, as a clue to techniques for solving your problem, is often a good thing to do, and may be a compromise between working solely on your problem and reading for general knowledge.