Some aspects of the Riemann-Hilbert correspondence

Abstract:

To any system of meromorphic linear differential equations on the Riemann sphere, we can associate a representation of the fundamental group of the punctured sphere, the so-called monodromy representation, measuring the dynamics of the solutions. Roughly speaking, the Riemann-Hilbert problem asks whether any representation can be obtained in this way. The answer to this problem highly depends on its precise statement in different contexts. We investigate the following angle : conveniently patching together local systems of differential equations with respect to the monodromy, we obtain a so called ''connection'' on a vector bundle, given by the global patching. What can be said about this vector bundle ?