BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Convex tilings by triangles and more Smillie, Peter


A triangulation of the sphere is combinatorially convex if each vertex is shared by no more than six triangles. In joint work with Philip Engel, we show that counted appropriately, the number of triangulations of the sphere with $2n$ triangles is the $n$th Fourier coefficient of a certain multiple of the Eisenstein series $E_{10}$. Our method is based on Thurston's description of triangulations as lattice points in a stratum of sextic differentials. It generalizes in a straightforward way to show that the number of convex tilings of a sphere by squares or by hexagons also form the coefficients of a modular form. As a consequence, we reproduce formulas for Masur-Veech volumes of certain strata of cubic, quartic, and sextic differentials. Time permitting, I will describe an approach to counting problems in strata of differentials of all orders.

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