Hilbert's twenty-first problem

The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, concerns the existence of a certain class of linear differential equations with specified singular points and monodromic group.

The original problem was stated as follows (English translation from 1902):

In fact it is more appropriate to speak not about differential equations but about linear systems of differential equations: in order to realise any monodromy by a differential equation one has to admit, in general, the presence of additional apparent singularities, i.e. singularities with trivial local monodromy. In more modern language, the (systems of) differential equations in question are those defined in the complex plane, less a few points, and with a regular singularity at those. A more strict version of the problem requires these singularities to be Fuchsian, i.e. poles of first order (logarithmic poles). A monodromy group is prescribed, by means of a finite-dimensional complex representation of the fundamental group of the complement in the Riemann sphere of those points, plus the point at infinity, up to equivalence. The fundamental group is actually a free group, on 'circuits' going once round each missing point, starting and ending at a given base point. The question is whether the mapping from these Fuchsian equations to classes of representations is surjective.

This problem is more commonly called the Riemann–Hilbert problem. There is now a modern (D-module and derived category) version, the 'Riemann–Hilbert correspondence' in all dimensions. The history of proofs involving a single complex variable is complicated. Josip Plemelj published a solution in 1908. This work was for a long time accepted as a definitive solution; there was work of G. D. Birkhoff in 1913 also, but the whole area, including work of Ludwig Schlesinger on isomonodromic deformations that would much later be revived in connection with soliton theory, went out of fashion. Plemelj (1964) wrote a monograph summing up his work. A few years later the Soviet mathematician Yuliy S. Il'yashenko and others started raising doubts about Plemelj's work. In fact, Plemelj correctly proves that any monodromy group can be realised by a regular linear system which is Fuchsian at all but one of the singular points. Plemelj's claim that the system can be made Fuchsian at the last point as well is wrong. (Il'yashenko has shown that if one of the monodromy operators is diagonalizable, then Plemelj's claim is true.)

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