Quick background: I'm a fourth year undergraduate entering graduate school next year. I am trying to identify areas of mathematical research in which there tends to be more emphasis on developing new theory as opposed to solving problems which can be formulated in terms of established theory.
Let me give two examples:
(Example 1) Suppose an analyst wants to study a particular PDE. In most cases, the PDE (which probably arose from phsyics or geometry) has been known for a long time. Moreover, the function spaces ($L^2, H^2,$ ect) which are usually used in formulating PDE problems have also been defined for the better part of a century. Now, there is a gigantic amount of room for innovation in modern PDE theory for developing new techniques. However, the basic questions (well posedness, unique continuation, ect) would have been intelligible sixty years ago.
(Example 2) A huge area of research in modern topology is the study of smooth 4-manifolds. The holy grail is the smooth 4-d Poincaré conjecture. Here again, there are incredibly sophisticated and beautiful tools being developed to answer questions about 4-manifolds. But the questions themselves are quite old. Fundamentally, we are trying to understand objects (namely smooth manifolds) whose definition has unchanged for the better part of a century.
My questions are:
A1 (Main question): Are there some areas of mathematics whose guiding questions would not be intelligible in terms of the theory we knew 20 years ago?
A2 (More subjective question): What are some areas where new theory is being developed as we speak, or where there seems to be a great need for new theory?
Remark: All mathematicians (at least all the one's I've spoken to) agree that good theory arises from good problems. However, I do get the feeling that some areas leave more room for theory than others, hence my question!