I'll begin by describing Witten's conjectures relating the KdV hierarchy to intersection numbers on the moduli space of stable curves. Then I'll explain how a similar set of conjectures arise in connection with moduli spaces of stable curves with boundary. An integrable system called open KdV takes the place of the KdV hierarchy. I will not assume any familiarity with intersection theory or stable curves. This is joint work with R. Pandharipande and R. Tessler.

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