Invariant Theory and Complexity Theory for Quiver Representations and Tensors

不变量理论研究的是在高维空间的对称性下保持不变的量。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来表示。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），图形和理论计算机科学中的问题的对称性。主要研究者积极参与研究生的培训，在接近本研究的领域。一个特别感兴趣的例子是同时左和右乘法的特殊线性群的n乘n矩阵的m元组的行动。主要研究者和他的合作者获得了基本多项式不变量的次数的多项式次数界，这一结果已被应用于获得非交换埃德蒙问题和非交换有理恒等式测试的确定性多项式时间算法。该项目的重点是张量的不变理论和构造性不变理论的新应用，如图同构问题和Brascamp-Lieb不等式。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.