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Friday, February 6, 2009 |
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The Asymptotic Enumeration of Maps |
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One area founded by W. T. Tutte in the 1960s is the enumeration of planar maps, graphs drawn in the plane with n edges. There has been a great deal of subsequent work and the area is still developing, especially when the graphs are embedded on a surface of genus g. Recently with Ed Bender and Jason Gao the asymptotic expression for the number of these maps on an orientable surface of genus g has been greatly simplified. More recently, with Nick Wormald, connections with the Painleve transcendent of type I and Gevrey Type 1 divergent series have been found. This work will be surveyed from a historical point of view. The results and methods are sometimes complicated so only the ideas and simpler results will be mentioned. |