Invariant Theory and Complexity Theory for Quiver Representations and Tensors

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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.