How to be a bird Freeman Dyson has famously characterized two styles of mathematics, that of the bird and that of the frog. I was asked recently the following question, which I don't know how to answer: If, in graduate school, you were raised by frogs, how do you go about becoming a bird?
(I'm sure the opposite question is also interesting.)
I feel like the question is really tough to answer, because what one "cuts one's teeth on" in graduate school so deeply biases you to a certain kind of question. My (tinny sounding) attempt at an answer was that one should simply look for problems one is attracted to and has the tools to try to solve. The person, I think, was more interested in knowing what the fundamental difference in "taste" and "style" is between the research sensibility of the bird and that of the frog....and how to move toward one style after having been trained in another...
 A: When people used to show sympathy towards Dyson, that he has not earned Nobel prize for QED (Quantum Electro-Dynamics), he replied (sorry I don't have reference) that to achieve such big awards, you need to find a big problem and then stick to it for at least ten years. This indeed is true. If you look at the lives of Field Medal winners, whether be S.T. Yau or Maryam Mirzakhani, they all had this thing in common, viz., they managed to get PhDs at a younger age, and won the medal for the work they had done in the last ten or so years (after completion of PhD). 
So, in order to be a bird, you got to be focussed, don't dilly dally from one theme to another, even if you understand all branches of mathematics. In 'Inner shape of space', Yau says that he took courses in almost every area of mathematics like probability, number theory etc. But he remained focussed on one area and gave not only mathematicians but also physicists, the Calabi-Yau manifolds. 
In conclusion, you got to follow the moderate path. As an example, these days you need to know a lot of number theory if you want to be an expert in algebraic geometry! So learn number theory, go deep, but don't get too excited. Hold your horses (do not switch between Poincare conjecture one week and Riemann hypothesis the other). Just study such excursions as a diligent and obedient student, then forget about the enjoyments of that field. Just use it as a tool in your work (which in our example is algebraic geometry).
Let's live moderate!
