The Riemann-Hilbert Problem and Integrable Systems, Volume 50, Number 11

is called Fuchsian if the N ×N coefficient matrix A(λ) is a rational function of λ whose singularities are simple poles. Each Fuchsian system generates, via analytic continuation of its fundamental solution Ψ (λ) along closed curves, a representation of the fundamental group of the punctured Riemann sphere (punctured at the poles of A(λ)) in the group of N ×N invertible matrices. This representation (or rather its conjugacy class) is called the monodromy group of equation (1), and it is the principal object of the theory of Fuchsian systems. The question of whether there always exists a Fuchsian system with given poles and a given monodromy group was included by Hilbert in his famous list as problem number twenty-one. The problem got the name “Riemann-Hilbert” for its obvious relation to the general idea of Riemann that an analytic (vector-valued) function could be completely defined by its singularities and monodromy properties. Subsequent developments put the RiemannHilbert problem into the context of analytic factorization of matrix-valued functions and brought to the area the methods of singular integral equations (Plemelj, 1908) and holomorphic vector bundles (Röhrl, 1957). This resulted eventually in a negative (!) solution, due to Bolibruch (1989), of the Riemann-Hilbert problem in its original setting and to a number of deep results (Bolibruch, Kostov) concerning a thorough analysis of relevant solvability conditions. We refer the reader to the book of Anosov and Bolibruch [2] for more on Hilbert’s twenty-first problem and the fascinating history of its solution (and for more details on the genesis of the name “Riemann-Hilbert”). Simultaneously, and to a great extent independently of the solution of the Riemann-Hilbert problem itself, a powerful analytic apparatus—the Riemann-Hilbert method—was developed for solving a vast variety of problems in pure and applied mathematics. The Riemann-Hilbert method reduces a particular problem to the reconstruction of an analytic function from jump conditions or, equivalently, to the analytic factorization of a given matrixor scalar-valued function defined on a curve. Following a tradition that developed in mathematical physics, it is these problems, and not just the original Fuchsian one, that we will call Riemann-Hilbert problems.1 In other words, we are adopting a point of view according to which the Riemann-Hilbert (monodromy) problem is formally treated as a special case (although an extremely important one) of a Riemann-Hilbert (factorization) problem. The latter is viewed as an analytic tool, but one whose implementation is not at all algorithmic and which might use quite sophisticated and Alexander R. Its is professor of mathematics at Indiana University–Purdue University at Indianapolis. His email address is [email protected].