Small polygons and toric codes

We describe two different approaches to making systematic classifications of plane lattice polygons, and recover the toric codes they generate, over small fields, where these match or exceed the best known minimum distance. This includes a [36,19,12]-code over F_7 whose minimum distance 12 exceeds that of all previously known codes.Comment: 9 pages, 4 tables, 3 figure

Databases of quantum periods for Fano manifolds

Fano manifolds are basic building blocks in geometry - they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in geometry, which has been open since the 1930s. One can think of this as building a Periodic Table for shapes. A recent breakthrough in Fano classification involves a technique from theoretical physics called Mirror Symmetry. From this perspective, a Fano manifold is encoded by a sequence of integers: the coefficients of a power series called the regularized quantum period. Progress to date has been hindered by the fact that quantum periods require specialist expertise to compute, and descriptions of known Fano manifolds and their regularized quantum periods are incomplete and scattered in the literature. We describe databases of regularized quantum periods for Fano manifolds in dimensions up to four. The databases in dimensions one, two, and three are complete; the database in dimension four will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed

The Geometry of T-Varieties

This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.Comment: 42 pages, 17 figures. v2: minor changes following the referee's suggestion