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| For Riemann–Hilbert factorization problems on the complex plane see [[Riemann–Hilbert]].
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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).
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| :'''''Proof of the existence of linear differential equations having a prescribed monodromic group'''''
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| :''In the theory of [[linear differential equation]]s 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 [[Mathematical singularity|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 [[local monodromy|linear substitution]]s. 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. {{harvs|txt|authorlink=Ludwig Schlesinger|first=L. |last=Schlesinger|year=1895}} has given this proof, based upon [[Henri Poincaré|Poincaré]]'s theory of the [[Fuchsian zeta-function]]s. 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.'' [http://aleph0.clarku.edu/~djoyce/hilbert/problems.html]
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| 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]].
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| 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. {{harvtxt|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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| Indeed {{harvs|txt|authorlink=Andrei Bolibrukh|first=Andrey A. |last=Bolibrukh|year=1990}} found a counterexample to Plemelj's statement.
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| This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind;
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| 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. {{harvs|txt|authorlink=Andrei Bolibrukh|first=Andrey A. |last=Bolibrukh|year=1992}} and independently {{harvs|txt|authorlink=Vladimir Petrov Kostov|first=Vladimir |last=Kostov|year=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 <math>n</math> with <math>p+1</math> poles which cannot be realised by Fuchsian systems equals <math>2(n-1)p</math> ({{harvs|txt|authorlink=Vladimir Petrov Kostov|first=Vladimir |last=Kostov|year=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 surface]]s. [[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.
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| ==References==
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| *{{Citation | last1=Anosov | first1=D. V. | last2=Bolibruch | first2=A. A. | title=The Riemann-Hilbert problem | publisher=Friedr. Vieweg & Sohn | location=Braunschweig | series=Aspects of Mathematics, E22 | isbn=978-3-528-06496-9 | mr=1276272 | year=1994}}
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| *{{Citation | last1=Bolibrukh | first1=A. A. | title=The Riemann-Hilbert problem | doi=10.1070/RM1990v045n02ABEH002350 | mr=1069347 | year=1990 | journal=Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | issn=0042-1316 | volume=45 | issue=2 | pages=3–47|language=Russian}}
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| *{{Citation | last1=Plemelj | first1=Josip | editor1-last=Radok. | editor1-first=J. R. M. | title=Problems in the sense of Riemann and Klein | url=http://books.google.com/books?id=f0urAAAAIAAJ | publisher=Interscience Publishers John Wiley & Sons Inc.\, New York-London-Sydney | series= Interscience Tracts in Pure and Applied Mathematics | mr=0174815 | year=1964 | volume=16}}
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| *{{Citation | last1=Bolibrukh | first1=A.A. | title=Sufficient conditions for the positive solvability of the Riemann-Hilbert problem. | journal=Matematicheskie Zametki | pages=9–19, 156 | language=Russian | journal=Mathematical Notes | year=1992 | volume=51 | issue=1-2 | pages=110–117 | language=English | mr=1165460}}
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| *{{Citation | last1=Kostov | first1=Vladimir Petrov | title=Fuchsian linear systems on <math>CP^1</math> and the Riemann-Hilbert problem | journal=Comptes Rendus de l'Académie des Sciences. Série I. Mathématique | year=1992 | volume=315 | issue=2 | pages=143–148 | mr=1197226}}
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| *{{Citation | last1=Schlesinger | first1=L. | title=Handbuch der Theorie der linearen Differentialgleichungen vol. 2, part 2, No. 366 | year=1895}}
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| ==External links==
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| * [http://www.gang.umass.edu/~kilian/mathesis/mathesis.html On the Riemann–Hilbert Problem] ([http://web.archive.org/web/20050305104624/http://www.gang.umass.edu/~kilian/mathesis/mathesis.html archive.org copy] [http://euclid.ucc.ie/pages/staff/mk/mathesis.pdf])
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| {{Hilbert's problems}}
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| [[Category:Hilbert's problems|#21]]
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| [[Category:Ordinary differential equations]]
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| [[Category:Algebraic curves]]
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