Hilbert's twenty-first problem

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The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, was phrased like this (English translation from 1902).

Proof of the existence of linear differential equations having a prescribed monodromic group

In the theory of linear differential equations with one independent variable z, I wish to indicate an important problem one which very likely Riemann himself may have had in mind. This problem is as follows: To show that there always exists a linear differential equation of the Fuchsian class, with given singular points and monodromic group. The problem requires the production of n functions of the variable z, regular throughout the complex z-plane except at the given singular points; at these points the functions may become infinite of only finite order, and when z describes circuits about these points the functions shall undergo the prescribed linear substitutions. The existence of such differential equations has been shown to be probable by counting the constants, but the rigorous proof has been obtained up to this time only in the particular case where the fundamental equations of the given substitutions have roots all of absolute magnitude unity. L. Schlesinger has given this proof, based upon Poincaré's theory of the Fuchsian zeta-functions. The theory of linear differential equations would evidently have a more finished appearance if the problem here sketched could be disposed of by some perfectly general method. [1].

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. It is now known in a modern (D-module and derived category) version 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 deformation that would much later be revived in connection with soliton theory, went thoroughly out of fashion. Plemelj produced a 1964 monograph Problems in the Sense of Riemann and Klein, (Pure and Applied Mathematics, no. 16, Interscience Publishers, New York) summing up his work. A few years later the Soviet mathematicians Yuliy S. Il'yashenko and others started raising doubts about Plemelj's work in detail. In fact, Plemelj's proof is correct when he claims that any monodromy group can be realised by a regular linear system which is Fuchsian at all singular points but one. Plemelj's claim that the system can be made Fuchsian at the last point as well is wrong, and in 1989 another Soviet mathematician Andrey A. Bolibrukh (1950-2003) brought forward a counterexample to Plemelj's exact formulation. Bolibrukh showed that for given poles certain monodromy groups can be realised by regular, but not by Fuchsian systems. This development had by then been somewhat overtaken by events: the Grothendieck school of algebraic geometry had become interested in the question of 'connections on curves', or in other words linear differential equations on Riemann surfaces. Pierre Deligne produced a new proof in modern language (the major point there is to say what 'Fuchsian' means for a system). With work by Rohrl, the case in one complex dimension was again covered.