Representation Theory Meets Computational Algebra and Complexity Theory

这个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)多项式方程系统，(2)计算机科学和计算复杂性。多项式方程组可以看作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，以及因此描述它们的模型，往往带有丰富的对称性。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理解该模型的几何学和实用性。该项目的第二个主题是矩阵乘法的复杂性(矩阵是一个由数字组成的矩形阵列)。寻找有效的矩阵相乘方法是计算机科学中被称为复杂性理论的一个子领域的主题。1968年，Strassen发现，被广泛使用的矩阵乘法算法被认为是最好的，但实际上并不是最优的。从那时起，在确定矩阵乘法的效率和确定Strassen算法可以改进的限度方面，人们进行了密集的研究。PI建议使用现代数学技术来解决这些问题。这个项目将通过为开源计算机代数系统Macaulay2开发新的软件，以及通过PI对扩大数学研究的参与的兴趣，产生相当广泛的影响。建议涉及几个主要主题：Weyman-Kempf几何技巧、合集和最小自由分辨率、割线变分和张量阶的研究。第一个目标是寻找新的例子并分析现有的例子，以扩展Weyman-Kempf几何技术来研究非正态簇。第二个目标是研究幂零轨道闭合和行列式增厚。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数的计算及其在矩阵乘法复杂性方面的应用。使用形变理论等现代代数几何的工具，PI将解决一些长期开放的猜想。该项目由数学科学部的代数和数论计划和既定的刺激竞争研究计划(EPSCoR)联合资助。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。

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.