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  "body": "\n## Problem\n\nLet $a_1, \\ldots, a_m$ be distinct points of the Riemann sphere, let $U$ be the sphere with these points removed, and fix $z_0 \\in U$.\nLet $\\rho \\colon \\pi_1(U, z_0) \\to \\mathrm{GL}_n(\\C)$ be a group homomorphism.\nDoes there exist a linear differential equation of Fuchsian class whose singular points are $a_1, \\ldots, a_m$ and whose [[monodromy representation|C8HD62AB6Q]] is $\\rho$?\n\n## Remarks\n\nIn the form used today, the question is asked for first-order systems $Y' = A(z) Y$ of Fuchsian type, $A(z) = \\sum_{i} \\frac{A_i}{z - a_i}$ with constant matrices $A_i$, which have simple poles only.\nIf one only asks for a system with [[regular singular points|H96HXG6CA0]], the answer is positive; this was shown by Plemelj in 1908.\nPlemelj's work was long taken as a positive solution for Fuchsian systems.\nIn 1989 Bolibrukh found a counterexample: a representation that is not the monodromy of any Fuchsian system with the given singular points.\nThe answer is positive when $\\rho$ is irreducible, a result of Bolibrukh and of Kostov.\nFor scalar equations of order $n$, extra apparent singular points are needed in general.\nPlemelj's approach reduces the question to a matrix [[Riemann–Hilbert problem|RY8YQ9JZF9]] on a curve through the singular points.\n\n## Explanations\n\n### L1 human\n\nA linear differential equation with a few singular points has a monodromy: a rule that says how its solutions get mixed when one walks around each singular point.\nHilbert asked for the converse.\nGiven the points and any prescribed mixing rule, is there a tame (Fuchsian) equation that produces exactly this rule?\nFor a long time the answer was believed to be yes, but in 1989 it turned out to be no in general.\n",
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