Factorization and vector bundles on curves

In the 1950's the analyst Grothendieck used Riemann-Hilbert factorization (of a transition function along the equator) to classify holomorphic vector bundles on the Riemann sphere. The higher genus cases are more complicated (obviously). In this informal talk (by an analyst to an audience of algebraic geometers) I would like to convey why an analyst might care about factorization, a few things which are known, and in particular how this relates to semistability.