Riemann–Hilbert correspondence

The Riemann–Hilbert correspondence is a correspondence between abstract algebra (specifically group theory) and mathematical analysis (specifically differential equations). Classically, David Hilbert posed his twenty-first problem, referencing earlier work by Bernhard Riemann. The basic idea of this problem can be illustrated with an example: the complex differential equation ${\displaystyle zf'(z)=1}$ has solutions ${\displaystyle f(z)=\log z+C}$, which is regular everywhere except at 0 and ${\displaystyle \infty }$ on the Riemann sphere. If we continue the function, following a loop around the origin, the value of the function changes by an integer multiple of ${\displaystyle 2\pi i}$. This phenomenon is called monodromy of the differential equation ${\displaystyle zf'(z)=1}$. The monodromy for this example thus corresponds to adding an integer multiple of ${\displaystyle 2\pi i}$, which is a representation of the fundamental group of the sphere punctured in two points. Hilbert's 21st problem asks whether every suitable monodromy representation arises from a linear differential equation with regular singularites.

Modern research on the Riemann–Hilbert correspondence generalizes this, from ordinary differential equations (on the Riemann sphere) to systems of partial differential equations on higher-dimensional complex manifolds, or higher genus Riemann surfaces. The problem is usually formulated as a correspondence between flat connections on algebraic vector bundles and representations of the fundamental group. The correspondence is between certain systems of partial differential equations (linear and having very special properties for their solutions) and possible monodromies of their solutions, and there are many generalizations and variants.

Such a result was proved for algebraic connections with regular singularities by Pierre Deligne (1970, generalizing existing work in the case of Riemann surfaces) and more generally for regular holonomic D-modules by Masaki Kashiwara (1980, 1984) and Zoghman Mebkhout (1980, 1984) independently. In the setting of nonabelian Hodge theory, the Riemann-Hilbert correspondence provides a complex analytic isomorphism between two of the three natural algebraic structures on the moduli spaces, and so is naturally viewed as a nonabelian analogue of the comparison isomorphism between De Rham cohomology and singular/Betti cohomology.