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From graph coloring to moduli of Riemann surfaces via random matrix theory


Speaker: Motohico Mulase, UC Davis
Location: 693 Kerr
Start time: Mon, Oct 29 2001, 4:10PM

The graph coloring problem is an old problem, first mentioned in a letter from DeMorgan to Hamilton in October 23, 1852, exactly 149 years ago. The problem of vertex coloring, including the famous 4-color problem, has been completely solved. But how about edge coloring? We also know that a color is a spectrum of light, so it takes arbitrary positive real numbers. Thus an edge can be colored with a continuous spectrum. This talk is aimed at explaining the recent developments in edge coloring. When we use a rational number to color an edge, we obtain an arithmetic algebraic curve, based on the work of Belyi and Grothendieck. If a real number is used, then an arbitrary Riemann surface appears, and it forms the basis of Witten-Kontsevich theory. (Bill Thurston told me that if a complex number is used as a "color," then hyperbolic 3-manifolds appear, but this one you have to ask Bill to explain.) These techniques, together with the analysis of random matrices, provide interesting geometric information of the moduli of Riemann surfaces.