Hyperplane Arrangements

这个项目是在计算交换代数和代数几何。 这些领域的一个中心方面是多变量多项式的研究-或多元多项式。 多元多项式在机械工程、机器人学、计算机辅助设计和数值偏微分方程等领域有着广泛的应用。 研究将集中在三个相互关联的领域，其中多元多项式发挥着重要作用。 第一种是线性子空间的构形，例如平面上的直线。 第二种是插值，它涉及用多项式模型拟合数据。 第三种是分段多项式函数，或样条，如贝塞尔样条常见的绘图程序。 这些领域中的每一个都有围绕组合学和几何学对相应代数对象的影响的主要未解决的问题。 该项目的一个反复出现的主题是使用刚度理论，它起源于结构工程，以阐明这种影响。 几条调查线是一个不同的社区之间的数值分析，刚性理论，交换代数和代数几何的合作努力。该项目包括为研究生提供培训机会。 该项目由代数与数论项目和刺激竞争研究的既定项目（EPSCoR）共同资助。该项目侧重于几何，组合学和代数之间的相互作用尚未得到很好理解的问题。 第一个目标是使用刚性理论来系统地产生具有固定组合的线布置，其导子模块具有基于几何形状的变化的结构，概括了齐格勒的示例。 理解这些例子进一步阐明了寺尾的猜想，该猜想提出了一个排列的自由度是组合的。第二个目标涉及到寻找一个反例的猜想在数值分析中使用的混合动力的符号和数值方法的基础上类似于刚性理论的技术。 这个长期存在的猜想提出了三角剖分上光滑三次样条空间的维度公式。 第三个目标是研究理想的符号幂和正则幂之间的渐近包容性，着眼于高度结构化的例子，最近与组合优化和线性规划的联系给了良好的进展前景。 尽管存在差异，但这三个重点领域允许进行丰富的技术交流。 例如，同调方法将超平面排列和样条连接到刚性，而麦考利逆系统将样条连接到符号幂。 这些项目是计算性的，将包括符号软件（Macaulay 2）和数值软件（Bertini）的代码开发。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project is in computational commutative algebra and algebraic geometry. A central aspect of these fields is the study of polynomials in many variables - or multivariate polynomials. Multivariate polynomials appear in a wide range of applications such as mechanical engineering, robotics, computer-aided design, and numerical partial differential equations. The research will be clustered in three interconnected areas where multivariate polynomials play an essential role. The first is configurations of linear subspaces, such as lines in the plane. The second is interpolation, which involves fitting data with a polynomial model. The third is piecewise polynomial functions, or splines, such as the Bezier splines common in drawing programs. Each of these fields have major unsolved conjectures revolving around the impact of combinatorics and geometry on corresponding algebraic objects. A recurring theme of the project is to use rigidity theory, which has its origins in structural engineering, to elucidate this impact. Several lines of inquiry are a collaborative effort among the disparate communities of numerical analysis, rigidity theory, commutative algebra, and algebraic geometry. The project includes training opportunities for graduate students. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).The project focuses on problems where the interactions between geometry, combinatorics, and algebra are not well-understood. A first goal is to use rigidity theory to systematically produce line arrangements with fixed combinatorics whose module of derivations has changing structure based on the geometry, generalizing an example of Ziegler. Understanding these examples sheds additional light on Terao's conjecture, which proposes that freeness of an arrangement is combinatorial. A second goal involves searching for a counterexample to a conjecture in numerical analysis using a hybrid of symbolic and numerical methods based on techniques analogous to rigidity theory. This conjecture, which is long-standing, proposes a formula for the dimension of the space of smooth cubic splines on triangulations. A third goal is to study asymptotic containments between symbolic and regular powers of ideals, with an eye toward highly structured examples where recent connections to combinatorial optimization and linear programming give good prospects for progress. Despite their differences, the three focus areas allow for a rich exchange of techniques. For example, homological methods connect hyperplane arrangements and splines to rigidity, while Macaulay inverse systems link splines to symbolic powers. The projects are computational in nature and will include the development of code in symbolic software (Macaulay2) and numerical software (Bertini).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.