My hunch is that it might be possible to create something like https://polymathprojects.org/ for mathematical physics and I'd like to know whether MathOverflow users can recommend some appropriate problems. As Abdelmalek Abdessalam pointed out in the comments below such a platform can potentially be used to develop research programs in mathematical physics that are suitable for polymath type collaborative work.

The gist of this project would be to use collaborate problem-solving to tackle open problems in mathematical physics. Timothy Gowers wrote a seminal blog post on the subject here: Is massively collaborative mathematics possible?

There's a good point made by a commenter below as to why not simply use the current polymath platform. Well, I think that it would be easier for such a platform to succeed and gain support among mathematical physicists if it was maintained by mathematical physicists and focused mainly on problems in the area of mathematical physics. This is not the current setup of the Polymath project.

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    $\begingroup$ I don't know why there was a downvote. Maybe the question could be reformulated as a list of research programs in mathphys, some of which may be suitable for polymath type collaborative work. I see on the right a link to a question "Open problems in PDEs..." so one could consider this one a duplicate. But the other question is not focused enough to be useful. It is better to concentrate on mathphys and not add PDEs and dynamical systems. $\endgroup$ – Abdelmalek Abdesselam Jan 25 '17 at 14:21
  • $\begingroup$ @AbdelmalekAbdesselam That's a very good idea. I'll add this to the question. $\endgroup$ – Aidan Rocke Jan 25 '17 at 15:39
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    $\begingroup$ Do physicists commonly reason by cases? Mathematicians do, and that allows polymath to succeed with division of labor. Consider the classification of finite simple groups, or the geometrization conjecture, or the recent "split or Johnson" approach to graph isomorphism. Are there analogs of this style of reasoning in mathematical physics? $\endgroup$ – Matt F. Jan 25 '17 at 15:54
  • $\begingroup$ @MattF. Yes. Reasoning by cases is employed in many areas of mathematical physics that I'm familiar with. In fact, I think this applies to any problem which is considered 'hard'. Researchers break it into many variants and then see whether a solution to one variant has more general implications. The gravitational n-body problem would be a good example. $\endgroup$ – Aidan Rocke Jan 25 '17 at 17:09
  • $\begingroup$ @MattF. I think I must also add a clarification. In most universities the mathematical physics group is contained within the math department so they are considered mathematicians. $\endgroup$ – Aidan Rocke Jan 25 '17 at 17:12

Polymath 7 tackled the hot spots conjecture(due to Jeffrey Rauch) in 2012:

Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary.

The goal of the project was to establish the conjecture for a large range of triangles and partial results were documented on the blog. I must note that the associated wiki page is also very useful as it lists different cases as well as different approaches.

I think this is a nice example of a non-trivial problem that can potentially be tackled on a collaborative problem solving platform focused on mathematical physics.

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    $\begingroup$ It's a good example to consider, and to compare with the 1999 list of open problems in mathematical physics at web.math.princeton.edu/~aizenman/OpenProblems.iamp. At first glance this question seems closer to the math end of the math-physics continuum than most of those, but maybe with more time I would appreciate the physical issues here or the mathematical issues of the others more. $\endgroup$ – Matt F. Jan 26 '17 at 23:33
  • $\begingroup$ @MattF. Some of the problems in that list would be very appropriate: web.math.princeton.edu/~aizenman/OpenProblems.iamp/… $\endgroup$ – Aidan Rocke Jan 28 '17 at 18:54
  • $\begingroup$ @MattF. However, I think the harder problem is not the lack of mathematical physics problems that are appropriate. The bigger issue might be to find an active mathematical physicist that's willing to organise this. It's a risk that involves cultural rather than technical obstacles. $\endgroup$ – Aidan Rocke Jan 28 '17 at 18:57

I recommend the following problem :

Proof the existence of the spectral gap for the fractional quantum Hall effect at least for a simplified model where Laughlin's wavefunction is the exact ground state.


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