Representation Theory Meets Computational Algebra and Complexity Theory

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.