Representation Theory Meets Computational Algebra and Complexity Theory

表示论遇见计算代数和复杂性理论

• 批准号: 2302375
• 负责人: Hang Huang
• 金额: $ 10.8万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302375&HistoricalAwards=false
• 关键词: Representation; Theory; Meets; Computational; Algebra

The goal of this project is to use mathematical tools to tackle computational and applied mathematical problems. The main themes are (1) Systems of Polynomial Equations, (2) Computer Science and Computational Complexity. Systems of polynomial equations can be thought of as describing the dependence relations between physical quantities in some models. The solution set of them describes the geometric shape of the model. Natural phenomena, and hence the model describing them, often come equipped with a rich symmetry. Hence it is natural to use symmetry-based methods to study them. The proposed research will lead to a better understanding of the geometry as well as the utility of the model. A second theme of the project is the complexity of matrix multiplication (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. In 1968, Strassen discovered that the widely used algorithm for matrix multiplication which was assumed to be the best possible, is in fact not optimal. Since then, there has been intense research in both determining just how efficiently matrices may be multiplied and determining the limits of how much Strassen's algorithm can be improved. The PI proposes to use modern mathematical techniques to tackle those problems. This project will have a substantial broader impact through the development of new software for the open-source computer algebra system Macaulay2, and through the PI’s interest in broadening participation in mathematical research.The proposal involves several main themes: Weyman-Kempf geometric techniques, syzygies and minimal free resolutions, secant varieties and the study of tensor ranks. The first goal is to find new examples and analyze existing examples to extend Weyman-Kempf geometric techniques to study non-normal varieties. The second goal is the study of nilpotent orbit closures and determinantal thickenings. The PI will use technical tools such as spectral sequence and Lie superalgebra representations to compute numerical and homological invariants of related varieties. The third topic is the computation of different tensor ranks and their application to matrix multiplication complexity. Using tools from modern algebraic geometry such as deformation theory, the PI will tackle a number of longstanding open conjectures.This project is jointly funded by the Algebra and Number Theory program in the Division of Mathematical sciences, and by the Established Program to Stimulate Competitive Research (EPSCoR).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.

该项目的目标是使用数学工具来解决计算和应用数学问题。主要主题是（1）多项式方程组，（2）计算机科学和计算复杂性。多项式方程组可以被认为是描述某些模型中物理量之间的依赖关系。它们的解集描述了模型的几何形状。自然现象，以及描述它们的模型，常常具有丰富的对称性。因此，很自然地使用基于语义的方法来研究它们。建议的研究将导致更好地理解的几何形状以及该模型的实用性。该项目的第二个主题是矩阵乘法的复杂性（矩阵是一个矩形数组）。寻找矩阵相乘的有效方法是计算机科学的一个子领域，称为复杂性理论。1968年，斯特拉森发现，广泛使用的矩阵乘法算法被认为是最好的，实际上不是最优的。从那时起，人们就在确定矩阵相乘的效率和确定斯特拉森算法可以改进的限度方面进行了深入的研究。PI建议使用现代数学技术来解决这些问题。该项目将通过为开源计算机代数系统Macaulay 2开发新软件，以及PI对扩大参与数学研究的兴趣，产生更广泛的影响。该项目涉及几个主要主题：Weyman-Kempf几何技术，syzygies和最小自由分辨率，割线品种和张量秩的研究。第一个目标是发现新的例子，并分析现有的例子，以扩展韦曼-肯普夫几何技术，研究非正规品种。第二个目标是研究幂零轨道闭包和行列式加厚。PI将使用谱序列和李超代数表示等技术工具来计算相关变种的数值和同调不变量。第三个主题是不同张量秩的计算及其在矩阵乘法复杂度中的应用。利用现代代数几何的工具，如变形理论，PI将解决一些长期存在的开放式问题。该项目由数学科学部的代数和数论项目共同资助，激励竞争研究计划（EPSCoR）该奖项反映了NSF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进行评估，被认为值得支持的搜索.