Mathematics Colloquia and Seminars

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An elliptic orbifold is a quotient of the plane by an orientation-preserving wallpaper group. Building on work of Eskin and Okounkov, we outline a proof that natural counts of ramified covers of an elliptic orbifold form the Fourier coefficients of a quasi-modular form of specified level and weight. This theorem allows us to answer questions such as: How many combinatorially inequivalent ways are there to tile an oriented surface by 1000 quadrilaterals, so that the valences of the vertices are 3, 3, 4, 4, ..., 4, 4, 5, 5? Declaring each tile to be metrically regular, square (resp. hexagon) tiled surfaces form a uniformly distributed lattice within a moduli space of quartic (resp. cubic) differentials with specified orders of zeroes and poles. Analyzing the asymptotic behavior as the number of tiles goes to infinity, one may verify that the Masur-Veech volumes of such moduli spaces are polynomial in pi. This proves a natural generalization of a conjecture of Kontsevich and Zorich to cubic and quartic differentials.