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I am thinking about some open problems in nonlinear functional analysis and I just wanted to know if there are any problems that have been solved by using Algebraic geometry techniques in these fields.

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    $\begingroup$ Not an answer, hence only a comment: probably not exactly what you are having in mind, but do Vinogradov's diffieties (see also ncatlab.org/nlab/show/diffiety) count? That aside, it's usually the other way around: methods from functional analysis help with problems in Algebraic Geometry. $\endgroup$
    – M.G.
    Commented Jul 25 at 14:00

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