We outline work in progress suggesting an algebro-geometric version of the Strominger-Yau-Zaslow conjecture. We define the notion of a toric degeneration, a special case of a maximally unipotent degeneration of Calabi-Yau manifolds. We then show how in this case the dual intersection complex has a natural structure of an affine manifold with singularities. If the degeneration is polarized, we also obtain an intersection complex, also an affine manifold with singularities, related by a discrete Legendre transform to the dual intersection complex. Finally, we introduce log structures as a way of reversing this construction: given an affine manifold with singularities with a suitable polyhedral decomposition, we can produce a degenerate Calabi-Yau variety along with a log structure. Hopefully, in interesting cases, this object will have a well-behaved deformation theory, allowing us to use the discrete Legendre transform to construct mirror pairs of Calabi-Yau manifolds. We also connect this approach to the topological form of the Strominger-Yau-Zaslow conjecture.

Abstract: We introduce and study the toric Hilbert scheme which parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal. 1.

Consider the moduli space M 0 of arrangements of n hyperplanes in general position in projective (r − 1)-space. When r = 2 the space has a compactification given by the moduli space of stable curves of genus 0 with n marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs (S, B) consisting of a variety S (possibly reducible) and a divisor B = B1 +.. + Bn, satisfying various additional conditions. We identify the closure of M 0 in the moduli space of stable pairs as Kapranov’s Hilbert quotient compactification of M 0, and give an explicit description of the pairs at the boundary. We also construct additional irreducible components of the moduli space of stable pairs.

Abstract. We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the Abel-Jacobi, Riemann-Roch and Riemann theta divisor theorems. 1.

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Abstract. We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc. 13:3 (2000), 611–637 not only in size, but also in that the associated toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5-dimensional polytopes with non-connected graph of triangulations.

Abstract. We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. This

1. Main definitions and results 3 2. General criteria 6 2.1. Seminormality and connectedness of isotropy groups 6

We describe algorithms which solve two classical problems in lattice geometry for any fixed dimension d: the lattice covering and the simultaneous lattice packing–covering problem. Both algorithms involve semidefinite programming and are based on Voronoi’s reduction theory for positive definite quadratic forms which describes all possible Delone triangulations of Z d. Our implementations verify all known results in dimensions d ≤ 5. Beyond that we attain complete lists of all locally optimal solutions for d = 5. For d = 6 our computations produce new best known covering as well as packing–covering

Let : P ! Q be an affine projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by (or -induced) and the fiber polytope of the projection: a polytope \Sigma(P; ) of dimension dim(P ) \Gamma dim(Q) whose faces are in correspondence with the coherent -induced subdivisions (or -coherent subdivisions). In this paper we investigate the structure of the poset of -induced refinements of a -induced subdivision. In particular, we define the refinement polytope associated to any -induced subdivision S, which is a generalization of the fiber polytope and shares most of its properties. As applications of the theory we prove that if a point configuration has non-regular subdivisions then it has non-regular triangulations and we provide simple proofs of the existence of non-regular subdivisions for many particular point configurations.