Moduli spaces parametrizing Betti local systems (ie, representations of the topological fundamental group) on a punctured projective line with fixed monodromy at the punctures underlie the Deligne-Simpson problem and Katz's theory of rigid local systems.

During this talk, we'll survey this story and then discuss ongoing work to construct and describe analogous moduli spaces in the context of p-adic cohomology, where overconvergent isocrystals take the place of Betti local systems.