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I am only a beginner in the field, but I would like to have at least a few examples of open problems in deformation theory to give me an idea of the actual status of the theory, and to help guide my learning as well.

I am interested both questions about specific deformation problems and the general theory (by which I mean problems starting with: let $F$ be a deformation functor...)

It would be nice to have an answer for every open problem posted, possibly with some details and references.

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  • $\begingroup$ Made CW on OP's request. $\endgroup$
    – Todd Trimble
    Commented Jul 7, 2016 at 2:06
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    $\begingroup$ Deformation of what? $\endgroup$
    – abx
    Commented Jul 7, 2016 at 6:13

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Deformation theory is of course a very very wide field and one can take many different points of view on it. Working in deformation quantization, i.e. formal associative deformations of algebras and their modules, one of the important questions these days is whether and how one can make a formal deformation convergent in some meaningful way. Ideally, one would like to relate the formal deformations (a la Gerstenhaber, ...) to more strict ones, perhaps in the formulation of Rieffel for $C^*$-algebras. Still very unspecific but quite nontrivial and interesting...

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