Combinatorial Commutative Algebra

这项建议使用代数和计算方法来研究代数、几何和组合学之间的问题。两个中心主题是环形变体和超平面排列。Toric簇的主要研究对象是等变Chow环、Toric簇的齐次坐标环和Toric码。这些问题是为了更好地理解组合学和几何(定义环面簇的扇形或定义除数的多面体)如何在Chow环、齐次坐标环和相关的环面码中表现。超平面排列的主要研究对象是对数一型模。特别是，Terao关于这个模的自由度是由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阶，以及它们与共振簇的联系。三个Toric项目中的两个具有重要的现实应用：Toric簇的Chow环就是样条环：这样的对象在数值分析(例如，在求解对许多应用数学至关重要的偏微分方程)和几何建模中是核心的。Toric码是在信号处理和数据压缩中使用的里德-所罗门和里德-穆勒码的推广。编码理论的进步可能导致在有噪声的通信信道上更有效地传输数据。将为NSF赞助的Macaulay2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，在数学物理中，圆环簇被广泛用作镜像对称性的测试用例。该软件将与研究生共同编写：该项目支持两名博士生50%的暑期研究。这个奖项由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.