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):
- 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 (1895) 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. 
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.)
Indeed Andrey A. Bolibrukh (1990) 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. (In 1990 he published the thorough study of the case of regular systems of size 3 exhibiting all situations when such counterexamples exists. In 1978 Dekkers had shown that for systems of size 2 Plemelj's claim is true. Andrey A. Bolibrukh (1992) and independently Vladimir Kostov (1992) showed that for any size, an irreducible monodromy group can be realised by a Fuchsian system. The codimension of the variety of monodromy groups of regular systems of size with poles which cannot be realised by Fuchsian systems equals (Vladimir Kostov (1992)).) 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 Deligne proved a precise Riemann–Hilbert correspondence in this general context (a major point being to say what 'Fuchsian' means). With work by Helmut Röhrl, the case in one complex dimension was again covered.
- Anosov, D. V.; Bolibruch, A. A. (1994), The Riemann-Hilbert problem, Aspects of Mathematics, E22, Braunschweig: Friedr. Vieweg & Sohn, doi:10.1007/978-3-322-92909-9, ISBN 978-3-528-06496-9, MR 1276272
- Bolibrukh, A. A. (1990), "The Riemann-Hilbert problem", Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk (in Russian), 45 (2): 3–47, doi:10.1070/RM1990v045n02ABEH002350, ISSN 0042-1316, MR 1069347
- Plemelj, Josip (1964), Radok., J. R. M. (ed.), Problems in the sense of Riemann and Klein, Interscience Tracts in Pure and Applied Mathematics, 16, New York-London-Sydney: Interscience Publishers John Wiley & Sons Inc., MR 0174815
- Bolibrukh, A.A. (1992), "Sufficient conditions for the positive solvability of the Riemann-Hilbert problem", Matematicheskie Zametki (in Russian): 9–19, 156 (translation in Math. Notes 51 (1–2) (1992) pp. 110–117), doi:10.1007/BF02102113, MR 1165460
- Kostov, Vladimir Petrov (1992), "Fuchsian linear systems on and the Riemann-Hilbert problem", Comptes Rendus de l'Académie des Sciences, Série I, 315 (2): 143–148, MR 1197226
- Schlesinger, L. (1895), Handbuch der Theorie der linearen Differentialgleichungen vol. 2, part 2, No. 366
- Katz, N.M. (1976), "An Overview of Deligne's work on Hilbert's Twenty-First Problem", Proceedings of Symposia in Pure Mathematics, 28