Abstract

A systematic bridge between the enumerative geometry of stable curves and integrable hierarchies of PDEs was build by Buryak; based on the geometry of the so-called double ramification cycles, one associates the integrable DR hierarchy to a collection of cohomology classes that respect i.a. the gluing of two curves along a node. This talk aims to discuss moduli spaces of stable curves and differentials, the double ramification cycles, and how these form the geometric input for two striking examples of this DR construction. Specifically, Buryak- Rossi-Zvonkine proved that the hierarchy associated to the Poincaré dual classes of the closures of the loci of curves that admit residueless meromorphic differentials with specified orders of zeroes and poles reduces to the paradigmal Kadomtsev-Petviashvili (KP) hierarchy, describing 2D shallow water waves, up to a coordinate transform. The proof demonstrates how the reconstruction of the integrals that count higher genus curves with certain conditions from only a couple geometric computations is understood as the reconstruction of the KP hierarchy from a few initial equations and the commutativity of the flows. Taking into account the spin-parity of the differentials, this method produces the BKP hierarchy, as demonstrated together with Klompenhouwer.

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