Combinatorial Commutative Algebra

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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.