Editing Monodromy (section)

==Differential equations in the complex domain==
One important application is to [[differential equation]]s, where a single solution may give further linearly independent solutions by [[analytic continuation]]. Linear differential equations defined in an open, connected set <math>S</math> in the complex plane have a monodromy group, which (more precisely) is a [[linear representation]] of the [[fundamental group]] of <math>S</math>, summarising all the analytic continuations round loops within <math>S</math>. The [[inverse problem]], of constructing the equation (with [[regular singularity|regular singularities]]), given a representation, is a [[Riemann–Hilbert problem]].

For a regular (and in particular Fuchsian) linear system one usually chooses as generators of the monodromy group the operators <math>M_j</math> corresponding to loops each of which circumvents just one of the poles of the system counterclockwise. If the indices <math>j</math> are chosen in such a way that they increase from <math>1</math> to <math>p+1</math> when one circumvents the base point clockwise, then the only relation between the generators is the equality <math>M_1\cdots M_{p+1}=\operatorname{id}</math>. The [[Deligne–Simpson problem]] is the following realisation problem: For which tuples of conjugacy classes in <math>\operatorname{GL}(n,\mathbb{C})</math> do there exist irreducible tuples of matrices <math>M_j</math> from these classes satisfying the above relation? The problem has been formulated by [[Pierre Deligne]] and [[Carlos Simpson]] was the first to obtain results towards its resolution. An additive version of the problem about residua of Fuchsian systems has been formulated and explored by [[Vladimir Kostov]]. The problem has been considered by other authors for matrix groups other than <math>\operatorname{GL}(n,\mathbb{C})</math> as well.<ref>{{Citation|author=V. P. Kostov|title=The Deligne–Simpson problem — a survey|journal=J. Algebra|volume=281|year=2004|issue=1|pages=83–108|mr=2091962|doi=10.1016/j.jalgebra.2004.07.013|arxiv=math/0206298|s2cid=119634752}} and the references therein.</ref>