For a smooth projective complex algebraic variety X, a theorem of Simpson classifies algebraic subvarieties of the moduli space of rank one flat connections on X, that remain algebraic as a subspace of the moduli of local systems via the Riemann-Hilbert correspondence. In this talk, I shall describe a new proof of Simpson's result using algebraization techniques originating in o-minimality. We shall also describe how the o-minimal approach adapts to prove a p-adic analog for Mumford curves.