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 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. 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.
History
This problem is more commonly called the Riemann–Hilbert problem. There is now a modern 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. 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. Indeed found a counterexample to Plemelj's statement. This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind; Bolibrukh showed that for a given pole configuration certain monodromy groups can be realised by regular, but not by Fuchsian systems. Parallel to this the Grothendieck school of algebraic geometry had become interested in questions of 'integrable connections on algebraic varieties', generalising the theory of linear differential equations on Riemann surfaces. Pierre Deligneproved a precise Riemann–Hilbert correspondence in this general context. With work by Helmut Röhrl, the case in one complex dimension was again covered.
