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Invariant Theory and Complexity Theory for Quiver Representations and Tensors

Quiver 表示和张量的不变理论和复杂性理论

基本信息

批准号：
2147769
负责人：
Harm Derksen
金额：
$ 29.3万
依托单位：
Northeastern University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2147769&HistoricalAwards=false
关键词：
Invariant Theory Complexity Quiver Representations

项目摘要

Invariant theory studies quantities that remain unchanged under symmetries in high dimensional spaces. For example, the distance of a point to the rotation axis is an invariant for the rotation symmetry. Invariant theory has many useful applications in physics, chemistry, and other areas of mathematics. A classical task in invariant theory is to find a finite list of fundamental invariants for a given group of symmetries, such that all invariants can be expressed in terms of the fundamental invariants. In this project, the principal investigator and his collaborators plan to find bounds for the size of fundamental invariants and apply these results to symmetries of high-dimensional arrays (also called tensors), graphs, and questions in theoretical computer science. The principal investigator is actively involved in the training of graduate students in fields close to this research.An example of particular interest is the action of simultaneous left and right multiplication of the special linear group on m-tuples of n by n matrices. The principal investigator and his collaborators obtained polynomial degree bounds for the degrees of fundamental polynomial invariants, and this result has been applied to get deterministic polynomial time algorithms for the noncommutative Edmond's problem and for noncommutative rational identity testing. This project focuses on invariant theory for tensors and new applications of constructive invariant theory, such as the graph isomorphism problem and Brascamp-Lieb inequalities.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.

不变量理论研究在高维空间对称下保持不变的量。例如，点到旋转轴的距离是旋转对称的不变量。不变量理论在物理、化学和其他数学领域有许多有用的应用。不变量理论中的一个经典任务是为给定的对称群找到一个有限的基本不变量列表，使得所有的不变量都可以用基本不变量表示。在这个项目中，首席研究员和他的合作者计划找到基本不变量大小的界限，并将这些结果应用于高维数组（也称为张量）、图的对称性和理论计算机科学中的问题。首席研究员积极参与与本研究相关领域的研究生培训。一个特别有趣的例子是在n × n矩阵的m元组上同时对特殊线性群进行左右乘法的作用。主要研究者及其合作者获得了基本多项式不变量阶的多项式阶界，并将这一结果应用于非交换Edmond问题和非交换有理恒等式检验的确定多项式时间算法。本项目主要研究张量的不变量理论和构造不变量理论的新应用，如图同构问题和Brascamp-Lieb不等式。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。