Invariant Theory, Tensors, and Applications

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In many areas of science and engineering, including signal processing, data mining, neuroscience, and chemometrics, data often appear in multidimensional arrays. Such multidimensional arrays are known as tensors. To understand the internal structure of the data at hand, it is useful to decompose a given tensor as a sum of simpler tensors. However, the classical method for such decompositions has flaws, such as numerical instability. One of the main goals of this research project is to develop the theoretical underpinning of a new method for tensor decomposition that will perform better in applications, particularly in biomedical signal processing. The project involves graduate and undergraduate students in the research.This research project studies tensors and invariant theory. The project aims to develop methods for lower and upper bounds for the rank of a tensor. The theory of convex decompositions of tensors will be developed and generalized. This new theoretical framework for tensor decompositions is an alternative to the low rank (PARAFAC) decomposition and is expected to behave better numerically. The investigator will study rings of (semi-)invariants for quivers and quivers with potentials. He will also explore tensor invariants and their relations to twisted commutative algebras.