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Friday, January 25, 2008 |
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The KP hierarchy, branched covers and triangulations |
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The KP hierarchy is a system of quadratic partial differential equations that arises in mathematical physics, and generalizes the KdV hierarchy. An explicit characterization of solutions is known in terms of Schur symmetric functions and Plucker equations, which are quadratic relations arising in algebraic geometry. We describe a general class of solutions to the KP hierarchy arising from this characterization. Special cases of this class are the generating series for various types of branched covers, and graphs and hypergraphs embedded in an orientable surface. We conclude with a particular application, to the enumeration of rooted triangulations in an orientable surface of arbitrary genus. In the latter case, we obtain a remarkably simple quadratic recurrence equation, for which we have no direct combinatorial construction, but that seems to yield an explicit asymptotic form. The presentation will be largely expository. |