Riemann Hilbert Correspondence
The Riemann Hilbert correspondence is a simplification for Hilbert’s twenty-first trouble to superior proportions in mathematics. The original setting for Riemann surfaces. The Riemann Hilbert is about the extinction of fixture equations and prescribed monodromy group. In high dimensions, Riemann turns up are replaced by difficult manifolds in dimension. And it was correspondence among cerate systems of partial derivative differential equations and possible monodromies of their solutions. Such a result was proved independently by Masaki Kashiwara (1980) and Zoghman Mebkhout (1980).
Riemann–Hilbert correspondence
In general there is a functor DR ask the Rham functor that is equivalence from the class of holonomic D-modules on X regular singularities to the category of perverted sheaves on X.
A D-module is something like a system of differential equations on X, and a local system on a subvariety is something like a description of possible monodromies, so this correspondence can be thought of as describing certain systems of differential equations in terms of the monodromies of their solutions.