Hyperplane Arrangements

这个项目是关于计算交换代数和代数几何的。这些领域的一个中心方面是研究多元多项式--或多元多项式。多元多项式在机械工程、机器人、计算机辅助设计、数值偏微分方程等领域有着广泛的应用。这项研究将集中在三个相互关联的领域，在这三个领域中，多元多项式发挥着至关重要的作用。第一种是线性子空间的配置，例如平面中的线。第二种是内插法，它涉及用多项式模型来拟合数据。第三种是分段多项式函数或样条线，如绘图程序中常见的Bezier样条线。这些领域中的每一个都有主要的未解决的猜想，围绕着组合学和几何学对相应的代数对象的影响。该项目的一个反复出现的主题是使用起源于结构工程的刚性理论来解释这种影响。有几条探索路线是数值分析、刚性理论、交换代数和代数几何等不同社区之间的合作努力。该项目包括为研究生提供培训机会。这个项目是由代数和数论项目和已建立的激励竞争研究计划(EPSCoR)共同资助的。该项目专注于几何、组合学和代数之间相互作用不被很好理解的问题。第一个目标是利用刚性理论系统地产生具有固定组合的线列，其导子模块的结构基于几何而变化，推广了Ziegler的一个例子。理解这些例子有助于进一步理解Terao的猜想，该猜想提出一种排列的自由性是组合的。第二个目标涉及在数值分析中使用基于类似刚性理论的技术的符号和数值方法的混合来寻找对猜想的反例。这一猜想由来已久，它给出了三角剖分上光滑三次样条曲线空间的一个维数公式。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，着眼于高度结构化的例子，在这些例子中，最近与组合优化和线性规划的联系给出了良好的发展前景。尽管存在差异，但这三个重点领域允许丰富的技术交流。例如，同调方法将超平面排列和样条线与刚性联系起来，而麦考利逆系统将样条线与符号幂联系起来。这些项目本质上是计算性的，将包括符号软件(Macaulay2)和数值软件(Bertini)的代码开发。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.