Combinatorial Commutative Algebra

本建议使用代数和计算方法来研究代数、几何和组合学的界面问题。两个中心主题是环面变化和超平面排列。在环型中研究的主要对象是等变周氏环、含有丰富除数的环型的齐次坐标环和环型码。这些问题是为了更好地理解组合学和几何（扇形定义环的变化，或多面体定义除数）如何在周氏环、齐次坐标环和相关的环码中表现出来。在超平面排列中研究的主要对象是对数形式的模。特别是，Terao关于该模块的自由度是组合决定的猜想是该领域的主要开放问题之一。PI还将研究基本群的LCS和Chen秩，以及它们与共振变体的联系。三个环面项目中的两个在现实世界中有重要的应用：环面变化的周氏环就是简单的样条环：这样的对象在数值分析（例如，在求解偏微分方程中，这对许多应用数学至关重要）和几何建模中是中心的。环面码是里德-所罗门码和里德-穆勒码在信号处理和数据压缩中的推广。编码理论的进步可能导致在嘈杂的通信信道上更有效地传输数据。软件将为NSF赞助的Macaulay2平台开发，并向公众开放，使许多不同领域的研究人员受益。例如，环面变体在数学物理中被广泛用作镜像对称的测试用例。该软件将与研究生合作编写：该项目支持两名博士生50%的夏季研究。PI还将制作一本关于超平面排列的最先进的书。该奖项由Alegrba和数论、组合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.