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.

该项目的目的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）多项式方程的系统，（2）计算机科学和计算复杂性。多项式方程系统可以被认为是描述某些模型中物理量之间的依赖关系。它们的解决方案集描述了模型的几何形状。自然现象，因此描述它们的模型通常配备了丰富的对称性。因此，使用基于对称性的方法来研究它们是很自然的。拟议的研究将使人们更好地了解该模型的几何形状和效用。该项目的第二个主题是矩阵乘法的复杂性（矩阵是数字的矩形阵列）。寻找有效的方法来繁殖材料是计算机科学子场的主题，称为复杂性理论。 1968年，斯特拉森（Strassen）发现，被认为是最好的矩阵乘法算法，实际上并不是最佳的。从那时起，就一直在确定如何有效地乘以乘以乘积的限制，并确定可以改善多少Strassen算法的限制。 PI建议使用现代数学技术来解决这些问题。 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.第一个目标是找到新的例子并分析现有示例，以扩展Weyman-Kempf几何技术来研究非正常品种。第二个目标是研究Nilpotent轨道闭合和确定的增厚。 PI将使用诸如光谱序列和谎言超级代表等技术工具来计算相关变化的数值和同源不变。第三个主题是计算不同张量等级及其在矩阵乘法复杂性中的应用。使用现代代数几何形状（例如变形理论）的工具，PI将解决许多长期的开放猜想。该项目由数学科学部的代数和数字理论计划共同资助，并通过既定的计划以及既定的计划来刺激竞争性研究（EPSCOR），以反映了NSF的构建范围。影响审查标准。