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The number of non-negatively curved triangulations of S^2
Peter Smillie (Harvard University)
Aula Beltrami - Giovedì 12 Gennaio 2017 h.15:30
Abstract. A triangulation of S^2 is combinatorially non-negatively curved if each vertex is shared by no more than six triangles. Thurston showed that non-negatively curved triangulations of S^2 correspond to orbits of vectors of positive norm in a lattice \Lambda \subset \mathbb{C}^{1,9} under the action of a group of isometries. We show that an appropriately weighted number of triangulations of S^2 with 2n triangles gives the coefficients of a modular form, and specifically that the number is
\frac{809}{2612138803200} \sigma_9(n)
where \sigma_9(n) = \sum_{d|n}d^9. This is joint work with Philip Engel.
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