ABSTRACT. We compute the local Gromov-Witten invariants of the "closed vertex", that is, a configuration of three P 1 's meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of P 3 and then to compute those invariants via the geometry of the Cremona transformation.

We compute the local Gromov-Witten invariants of certain configurations of rational curves in a Calabi-Yau threefold. These configurations are connected subcurves of the "minimal trivalent configuration", which is a particular tree of ‫ސ‬ 1 's with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov-Witten invariants of a blowup of ‫ސ‬ 3 at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov-Witten invariants using the mathematical and physical theories of the topological vertex. In particular, we provide further evidence equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex.14N35; 53D45

The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is S4. There are three pairs of three-dimensional reflexive polytopes with this symmetry group, up to isomorphism. We identify a natural one-parameter family of K3 surfaces corresponding to each of these pairs, show that S4 acts symplectically on members of these families, and show that a general K3 surface in each family has Picard rank 19. The properties of two of these families have been analyzed in the literature using other methods. We compute the Picard-Fuchs equation for the third Picard rank 19 family by extending the Griffiths-Dwork technique for computing Picard-Fuchs equations to the case of semiample hypersurfaces in toric varieties. The holomorphic solutions to our Picard-Fuchs equation exhibit modularity properties known as "Mirror Moonshine"; we relate these properties to the geometric structure of our family. * We thank Andrey Novoseltsev for thoughtful discussion of computational techniques, and Charles Doran for inspirational conversations.

We prove that if $P(x)$ is a polynomial with nonnegative nondecreasing coefficients and $n$ is a positive integer, then $P(x+n)$ is unimodal. Applications and open problems are presented.

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