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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!

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  • $\begingroup$ I'm having trouble understanding what you mean by "new theory", since it seems that under any reasonable definition the answer is "all areas of mathematics are in the need for new theory". For instance, would you consider that the fact that all elliptic curves over $\mathbb Q$ are modular "old theory" since one of the guiding questions (Fermat's Last Theorem) was intelligible to mathematics centuries ago? What about Connes' approach to the Riemann hypothesis (an old question) in terms of operator algebras? $\endgroup$ – Santiago Canez Apr 25 '14 at 18:07
  • $\begingroup$ I guess it's a little bit like the distinction between lemma and theorem (not always obvious of course). Although I don't know very much about FLT, I gather that one of the remarkable things about the proof is that it relied on a lot of modern theory, and generated theory of its own. But I don't think this is true in general. I can think of plenty of problems whose resolution requires the introduction of remarkable techniques, but which did not generate a new class of mathematical objects worthy of intrinsic study. In fact, I would say that in some fields (like PDEs) this is the norm. $\endgroup$ – user142700 Apr 25 '14 at 18:18
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Tropical algebraic geometry is being developed as we speak though I can't vouch for any "great need" as you put it.

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Mirror symmetry almost certainly fits the bill; it didn't exist as a subject until the 1990's. More generally, my sense is that symplectic geometry/topology is developing theory quite rapidly.


But I don't think I understand your Example 2. Sure, much of the motivation behind $4$-manifold theory comes from classical questions. But many of these classical questions are quite hard, hence the need for sophisticated technology (as you mention).

All of this is to say that many of the "guiding questions" today are about these new invariants themselves, and likely would not be understood 20 (okay, maybe 30) years ago. In other words, my sense is that questions about these invariants have largely replaced classical questions in terms of what is actually done day-to-day.

If anything, it seems to me (a complete beginner) that new invariants are being proposed or discovered quite rapidly, and it seems like there's still a lot of space for new ones.

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  • $\begingroup$ Thanks for the answer. I think you make a good point regarding example 2. $\endgroup$ – user142700 Apr 30 '14 at 5:13

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