Question

I think it would be interesting to have a list of important unsolved problems in mathematics. What are the important (interesting) problems in your field of work? It would be especially nice, to have a list of "non-mainstream-problems" in the miscellaneous areas of mathematics.

Answer 1

The big open problem around where I work (D-modules) is the Jacobian Conjecture. This states that if you have an algebraic map f from C^n to C^n whose jacobian determinant is a non-zero constant, then f has an inverse.

The reason its related to D-modules is because is equivalent to Dixmier's conjecture, which states that every non-zero endomorphism of a Weyl algebra (the ring of polynomial differential operators in n variables) is an automorphism.

Its important to know about, not so that we can try to prove it, but so that we know what simple sounding things are hard. Several times, I have played with a simple sounding proposition for an hour or two, before realizing that it is equivalent to Dixmier's conjecture. Hence, I regard the conjecture as a "Here There Be Monsters" warning.

Answer 2

In some fields, like analytic number theory, new methods (and improvements in the known ones) are most important. For any particular open problem, and a powerful new method that solves it, there are usually several other open problems that also can be attacked by the new method. In such a situation it is hard to say that some particular one of those problems is peculiarly important. (Of course, analytic number theory does have a peculiarly important problem)

Answer 3

There's an excellent internet resource for exactly this question called the "Open Problem Garden" moderated by Matt DeVos and Robert Samal. Currently it's got a bit of a graph theory/combinatorics bent, but it's well set up for people to post and read open questions in all subjects.