Computational Algebra and Applications

许多现实世界的问题都有重要的数学成分。这个项目使用计算代数来解决这类问题。主要主题是：(1)数据分析；(2)图形学、可视化、几何建模；(3)计算机科学和计算复杂性。数据分析中的一个主要问题是从大量数据集中提取意义：想象一个密集的移动点云。一种方法是在某一时刻冻结云，然后增加单个点的大小，直到附近的组合并。拓扑数据分析使用复杂的数学工具来研究这个问题，并导致对视觉皮层活动，癌症病理和病毒进化的见解。从波音（Boeing）到皮克斯（Pixar），从分析湍流流动（飞机机翼上的空气通道）到精确模拟和虚拟现实，各种公司都在使用计算机图形学和几何建模。关键的数学工具是样条曲线，它提供了一种将局部片段组合成连贯的大图的方法。这个项目的第三个主题是动态过程的建模，这通常归结为多次乘以（大量）矩阵（矩阵是一个矩形的数字数组）。寻找矩阵乘法的有效方法是计算机科学的一个分支领域，即复杂性理论的主题。该项目建立在PI先前的工作基础上，使用称为永久值的代数对象来提供对问题的洞察。在爱荷华州立大学任职期间，PI与爱荷华州立退伍军人中心合作，为退伍军人学生提供额外的数学辅导和支持，包括在每个学期开始时进行为期一天的密集数学训练营。在这个项目中，他将在奥本大学开发类似的项目，第一个新兵训练营计划于2021年秋季举行。该提案围绕着几个主要主题：永久、多拓扑参数化、持久同调、近似理论和样条。第一组主题由同调方法和简单复合体统一，第二组主题由代数和离散几何的相互作用统一；研究对象是计算数学的核心参与者。这些问题有三个特点：它们在现实世界的应用中很重要；微妙的几何或组合数据以代数方式反映；它们在计算上是可处理的。该项目将使用计算代数的力量来发现重要的应用不变量与几何、组合学和代数之间的隐藏联系。在代数方面，对应用中的关键不变量（例如，多边形细分的样条空间的维数，Wachspress参数曲面的方程，持续同调的支持轨迹）进行了优雅的描述。这些问题的具体和计算方面意味着它们是研究生和博士后参与的理想选择，并使进展前景良好。除了科学进步之外，该项目还将与学生合作开发专门的软件，并将其集成到nsf赞助的Macaulay2软件包中。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many real world problems have a significant mathematical component. This project uses computational algebra to attack such problems. The main themes are: (1) Data Analysis, (2) Graphics, visualization, geometric modeling, (3) Computer science and computational complexity. A main problem in Data Analysis is to extract meaning from a massive dataset: imagine a dense, cloud of moving points. One approach is to freeze the cloud at a moment in time, and then increase the size of the individual points until nearby groups coalesce. Topological Data Analysis uses sophisticated mathematical tools to study the problem, and has led to insights into visual cortex activity, cancer pathology, and viral evolution. Companies from Boeing to Pixar use computer graphics and geometric modeling in applications ranging from analyzing turbulent fluid flow (air passage over a plane's wing) to accurate simulations and virtual reality. Key mathematical tools are splines, which provide a way of assembling a coherent big picture from local pieces. A third theme of this project is the modeling of dynamic processes, which often comes down to multiplying (massive) matrices many times (a matrix is a rectangular array of numbers). Finding efficient ways to multiply matrices is the topic of a subfield of Computer Science known as complexity theory. The project builds on prior work of the PI using algebraic objects known as permanents to provide insight into the problem. While at his previous position at Iowa State, the PI partnered with the Iowa State Veterans Center to provide extra math tutoring and support for student veterans, including an intensive day long Math Boot Camp at the start of each semester. In this project, he will develop similar programs at Auburn, with the first boot camp planned for fall 2021.The proposal revolves around a few main themes: permanents, polytopal parameterizations, and persistent homology, approximation theory and splines. The first group of topics is unified by homological methods and simplicial complexes, and the second by the interplay of algebra and discrete geometry; the objects of investigation are central players in computational mathematics. The problems share three traits: they are of importance in real world applications; subtle geometric or combinatorial data is reflected algebraically; and they are computationally tractable. The project will use the power of computational algebra to discover hidden connections between important applied invariants and geometry, combinatorics, and algebra. There are elegant characterizations of key invariants for applications (e.g. dimension of the spline space for a polytopal subdivision, equations for Wachspress parametric surfaces, the support locus of persistent homology) in terms of algebra. The concrete and computational aspect of the problems means they are ideal for involving graduate students and postdocs, and makes prospects for progress excellent. In addition to scientific advances, the project will result in development of specialized software, to be coauthored with students and integrated into the NSF-sponsored Macaulay2 software package.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.