Combinatorial Commutative Algebra

这个建议使用代数和计算方法来研究代数，几何和组合学的接口问题。两个中心主题是复曲面的品种和超平面安排。复曲面簇的主要研究对象是等变Chow环、复曲面簇的齐次坐标环和复曲面码。这些问题是为了更好地理解组合学和几何学（定义复曲面簇的扇形，或定义除数的多面体）如何在周环，齐次坐标环和相关的复曲面代码中表现出来。超平面排列的主要研究对象是对数1型模。特别是，寺尾的猜想，该模块的自由度是combinatorially确定的是在该领域的主要开放问题之一。PI还将研究LCS和陈秩的基本群，以及它们与共振品种的联系。三个复曲面项目中的两个具有重要的真实的世界应用：复曲面品种的周环简单地是样条环：这样的对象是中心的数值分析（例如，在解决偏微分方程，这是至关重要的大部分应用数学），并在几何建模。复曲面码是用于信号处理和数据压缩的Reed-Solomon和Reed-Muller码的推广。编码理论的进步可能会导致在嘈杂的通信信道上更有效地传输数据。将为NSF赞助的Macaulay 2平台开发软件，并公开提供，使许多不同领域的研究人员受益。例如，复曲面变体被广泛用作数学物理中镜像对称性的测试案例。该软件将与研究生合著：该项目支持两个博士的50%的夏季研究。学生PI还将制作一本关于超平面排列的最新书籍。该奖项由Alegrba和Nnmber理论，组合数学和应用数学项目共同资助。

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.