This proposal uses algebraic and computational methods to study problems at the interface of algebra, geometry, and combinatorics. The two central themes are toric varieties and hyperplane arrangements. The main objects studied in toric varieties are the equivariant Chow ring, the homogenous coordinate ring of a toric variety embedded by a very ample divisor, and toric codes. The questions are to better understand how combinatorics and geometry (of the fan defining the toric variety, or of the polytope defining the divisor) manifests in the Chow ring, in the homogeneous coordinate ring, and in the associated toric code. The main object studied in hyperplane arrangements is the module of logarithmic one forms. In particular, Terao's conjecture that the freeness of this module is combinatorially determined is one of the main open questions in the field. The PI will also study the LCS and Chen ranks of the fundamental group, and their connection to resonance varieties.
Two of the three toric projects have significant real world applications: the Chow ring of a toric variety is simply the ring of splines: such objects are central in numerical analysis (for example, in solving partial differential equations, which are crucial to much of applied mathematics), and in geometric modelling. Toric codes are a generalization of the Reed-Solomon and Reed-Muller codes used in signal processing and data compression. Advances in coding theory could lead to more efficient transmission of data over noisy communication channels. Software will be developed for the NSF sponsored Macaulay2 platform, and made publicly available, benefitting researchers in many different areas. For example, toric varieties are widely used as test cases for mirror symmetry in mathematical physics. This software will be coauthored with graduate students: the project supports 50% summer research for two Ph.D. students. The PI will also produce a state of the art book on hyperplane arrangements.
This award is cofunded by Alegrba and Nnmber Theory, Combinatorics, and Applied Mathematics programs.
" involves both pure and applied mathematics. Put simply, the objects of interest have a geometric flavor, for example, a collection of lines in the plane or a collection of planes in three-space. There were two distinct types of object studied; the class of hyperplane arrangements (described above), and toric varieties. While the objects are of interest in pure mathematics, they also appear in a myriad of applications, ranging from physics (string theory) to electrical engineering (signal transmission and clean sound on a CD player) to computer science (computer animation in movies like Toy Story). The project produced over a dozen papers with results of interest in pure mathematics. Three of the papers have serious applied implications for the field of computer vision and geometric modeling. One of the results involves Wachspress surfaces. In modeling, it is often useful to deform a planar shape by dragging a single point to a new position. The basic idea is to inscribe a shape inside a control polygon, and then move the shape by moving the vertices of the polygon. It turns out that mechanism to do this involves special coordinates known as barycentric coordinates, and one outcome of the project was a paper describing the relations that barycentric coordinates satisfy. Work is currently underway to generalize these results from two dimensions to three dimensions, which is also of importance in computer science. In addition to the scientific outcomes, the project resulted in development of human resources; one undergraduate, four graduate students, and one postdoctoral researcher were involved in the investigations, writing code and performing experiments. The PI also organized seven workshops/conferences during the course of the grant, with several of the workshops targeted specifically at graduate students and young researchers (for example, in summer 2011 the PI and his collaborator D. Cox taught an intensive 2 week workshop for graduate students on toric varieties; the PI gave 12 lectures over a 10 day span).
