Important (interesting) unsolved problems [closed]

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.

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There are already at least two good resources for this: Wikipedia and the open problem garden. I think we're straying into encyclopedia territory here. – Qiaochu Yuan Oct 25 2009 at 13:36
Thanks for pointing that out. Do you think, i should delete this question? – Spinorbundle Oct 25 2009 at 17:50
Let's not delete this, but it's possible we should close it. Please have a look at the discussion at meta.mathoverflow.net/discussion/8/… about open questions. I don't think this question is good, as asked -- it would be better to be more specific, and ask about important unsolved problems in particular fields. There are just way too many problems that fit the current criterion. If anyone else agrees, let's close this one and try again with something more specific. – Scott Morrison Oct 25 2009 at 18:27
Does the Stack Overflow software give you some finite set of reasons for closure to choose among? 'Cause this isn't "not a real question"... – Alex Fink Oct 25 2009 at 23:24
@Alex: yes. The options are "off topic", "exact duplicate", "subjective or argumentative", "spam", "blatently offensive", "no longer relevant", and "too localized". Which would you have picked? – Anton Geraschenko Oct 26 2009 at 4:01
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closed as not a real question by Anton Geraschenko♦♦Oct 25 2009 at 18:38

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.

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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)

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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.

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 "a bit of a graph theory/combinatorics bent" is a bit of an understatement :-) There's 135 Graph Theory problems listed, more than Number Theory, Analysis, Algebra, Topology AND Combinatorics combined... Very interesting attempt though! I'll try to do my part to make it better. – Alon Amit Oct 25 2009 at 17:33 I just checked out the "Open Problem Garden" and I think the above comment is an understatement. I think that maybe this question is a valid one for MathOverFlow. – Bart Snapp Jun 6 2010 at 22:28