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Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please. … Meaning of: long open The problem should occur in the literature or have a solid history as folklore. …

The Busemann-Petty problem (posed in 1956) has an interesting history. … For three years everyone believed the problem had been solved, but in 1997 Alexander Koldobsky (who was working on completely different problems) provided a new Fourier analytic approach to convex bodies …

He suggested the problems and helped in clarifying the solutions. Without him the work would not have started, progressed or ended. … Which leaves open the question of what is the author's contribution to the paper. …

But we have just learned that this problem becomes less serious as we take $p$-power roots. … By a theorem of Huber, we can find a small open neighborhood $\tilde{X}$ of $X$ with the same étale cohomology. …

At the moment, I'm working on a number of problems related to resonance counting. … I encounter many problems when it comes to research, such as staying up to date on current topics, finding open problems which suit my skills and interests, and finding papers on topics I need to study …

The category of hyperstonean topological spaces and open continuous maps. The category of hyperstonean locales and open maps. The category of measurable locales (and arbitrary maps of locales). … if the fibers can be nonseparable, and I do not know how to fix this problem in the point-set framework. …

, Physics: Brachistochrone problem (1696), Ising model (1925), The harmonic oscillator,(?) … 1854) punctured open set in C^n (Hartog's theorem *1906 ?) …

So I suppose what I'm after is problems where essentially the only difficulty is the need for the clever and unexpected idea. … I.e., I'm looking for problems that are very good challenge problems for working out how a computer might do mathematics. …

It is indeed an open problem, as Misha said. But here is a solution. In E. Golod, Some problems of Burnside type. 1968 Proc. Internat. Congr. Math. (Moscow, 1966) pp. 284-289. …

To counter that, we would want to use one of the other reasons, such as the "Having a proof gives more insight into the problem" justification. It would be great to see some good examples of that. … Further addition: It occurs to me that my question as phrased is open to misinterpretation, so I would like to have another go at asking it. …

Part of the problem is that this is not an issue in the Hermitian case, which is usually the case one is most frequently exposed to.) … Zorn's lemma gives a linear right inverse; the open mapping theorem gives a bounded right inverse. But getting a right inverse that is simultaneously bounded and linear is not always possible!) …

In my early work on the period-index problem I actually reached a contradiction via this mistake and remained there for several days until Cathy O'Neil set me straight. … Every finite index subgroup of a profinite group is open. …

Background Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. … The polymath projects so far consisted of an attempt to solve a specific open problem but some of the proposals were of different nature. …

So, although I guess the answer is no, I'll ask my official question in an open way : Is $\pi_1(X)$ free for $g\geq 2$ ? Edit: Users have now brilliantly solved the problem in multiple ways. …

It is still an open problem whether they are equal or not. The first proofs of the crossing lemma did not make the distinction. … Székely (2016): Turán’s Brick Factory Problem: The Status of the Conjectures of Zarankiewicz and Hill. In: R. Gera et al. (eds.)(2016): Graph Theory—favorite conjectures and open problems. 1.) …