A Riemann surface appears in many different guises in mathematics, for example, as a branched cover of the Riemann sphere, an algebraic subset of a projective space, or a complex analytic 1-manifold. What is the relationship between various representations of the same Riemann surface? In the first part of my talk, I will describe a conjectural answer to one aspect of this question, due to Mark Green. In the second part, I will talk about ribbons. Ribbons are a particular kind of non-reduced schemes—spaces that carry “infinitesimal functions.” I will explain how studying these seemingly strange objects helps us understand properties of regular Riemann surfaces relevant for Green’s conjecture.

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