differential equations, where a single solution may give further linearly independent solutions by analytic continuation. Linear differential equations defined in an open, connected set S in the complex plane have a monodromy group, which (more precisely) is a linear representation of the fundamental group of S, summarising all the analytic continuations round loops within S. The inverse problem, of constructing the equation (with regular singularities), given a representation, is called the Riemann–Hilbert problem. For a regular (and in particular Fuchsian) linear system one usually chooses as generators of the monodromy group the operators Mj corresponding to loops each of which circumvents just one of the poles of the system counterclockwise. If the indices j are chosen in such a way that they increase from 1 to p + 1 when one circumvents the base point clockwise, then the only relation between the generators is the equality . The Deligne–Simpson problem is the following realisation problem: For which tuples of conjugacy classes in GL(n, C) do there exist irreducible tuples of matrices Mj 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 GL(n, C) as well. Topological and geometric aspects In the case of a covering map, we look at it as a special case of a fibration, and use the homotopy lifting property to "follow" paths on the base space X (we assume it path-connected for simplicity) as they are lifted up into the cover C. If we follow round a loop based at x in X, which we lift to start at c above x, we'll end at some c* again above x; it is quite possible that c ≠ c*, and to code this one considers the action of the fundamental group 1(X, x) as a permutation group on the set of all c, as a monodromy group in this context. In differential geometry, an analogous role is played by parallel transport. In a principal bundle B over a smooth manifold M, a connection allows "horizontal" movement from fibers above m in M to adjacent ones. The effect when applied to loops based at m is to define