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 "nonmainstreamproblems" in the miscellaneous areas of mathematics.

closed as not a real question by Anton Geraschenko Oct 25 '09 at 18:38
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The big open problem around where I work (Dmodules) 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 nonzero constant, then f has an inverse. The reason its related to Dmodules is because is equivalent to Dixmier's conjecture, which states that every nonzero 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. 


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. 


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) 

