# Have any MO-hard problems ascended to “real” open problems in their fields?

Have any questions first proposed on Mathoverflow attracted enough interest from experts in their field that solving them would be considered a significant advance?

I don't want to count problems that are known (or strongly suspected) to be at least as hard as some previously described problem, unless the version original to Mathoverflow is believed by experts to be a better formulation of the problem.

Of course, as Mathoverflow has been around for less than 10 years, no problem original to Mathoverflow can possibly be a long-standing open problem in its field. But I don't see any reason a question asked on MO can't be among the most interesting questions asked in the last ten years.

• How do you measure how interesting a problem is? -- Is it the total amount of monetary prizes one can win for a solution, the number of times it is mentioned in the literature, the number of papers which would be "erased" by solving it? – Stefan Kohl Sep 1 '17 at 20:01
• If such were the case, I think it would have shown up by now as a success story in another meta thread. I think we have to wait longer to see the kind of behaviour you mention. Gerhard "Asked Several 'Not-Yet-Famous' Original Problems" Paseman, 2017.09.01. – Gerhard Paseman Sep 1 '17 at 20:06
• @StefanKohl Ideally by expert opinion. For objective measure, I'm looking for literature mentions. For instance, people sometimes publish lists of open problems. If any MO-original problem showed up on one of those lists, that would count. – Will Sawin Sep 1 '17 at 20:24
• @StefanKohl For your third measurement, there actually are some examples. There have been MO questions asked which motivated research, where the research only partially solved the problem. Hence these papers would be erased by a full solution. For your first measurement, I doubt there are any examples, but one could easily be created. – Will Sawin Sep 1 '17 at 20:25
• @GerhardPaseman My thinking was that perhaps coming up with a new open question is not as big a "success story" as writing a paper, and so people might not have mentioned such a thing on that thread. But perhaps I am wrong. If so, maybe I should ask about questions which aren't yet famous, but their authors think should be famous. – Will Sawin Sep 1 '17 at 20:31
• @WillSawin: As to problems in published lists of open problems -- there are e.g. a couple of questions of mine in the Kourovka Notebook which I posed first on MO. – Stefan Kohl Sep 1 '17 at 21:50
• I'll just add that "another meta thread" which @GerhardPaseman refers to in the above comment is (most likely) this one: Best of MathOverflow. (Other questions which are linked there might be of interest, too.) – Martin Sleziak Sep 2 '17 at 6:07
• @StefanKohl Well that is the best example so far. – Will Sawin Sep 2 '17 at 6:46
• See also MO-Hard Questions. – Joseph O'Rourke Sep 2 '17 at 12:33
• I'm voting to close this question as off-topic. – Joseph Van Name Sep 5 '17 at 13:24
• I'm voting to keep this question open as on-topic. – Joonas Ilmavirta Sep 10 '17 at 11:20

I am not sure if the following example counts, since the question (by Ozawa) was asked specifically with a view to solving an existing open problem. Nevertheless: the answer to Szőkefalvi-Nagy's unitarizability theorem in the Calkin algebra? turns out to be positive; yet a related problem, namely "is every bounded representation of a discrete abelian group inside the Calkin algebra unitarizable?" turns out to have a negative answer, and -- by the reasoning which Ozawa outlined in the MO post -- one can use this to construct the first known example of an amenable subalgebra of B(H) that is not isomorphic to the underlying Banach algebra of any $${\rm C}^*$$-algebra. This example is due originally to Farah and Ozawa; a simplified version appears in a paper of myself and those two.