Dr. Clifford Smyth, an Associate Professor in the Department of Mathematics and Statistics, has received a $42,000 grant from the Simons Foundation for his proposal, “Determinantal Identities and the Rational Non-negativity Problem.”
The abstract below provides additional info on this project.
The proposed research is on a series of combinatorial question and a novel research program in computational mathematics merging the established field of certification of 0-dimensional algebraic varieties with differential equation prediction and questions of connectivity.
The combinatorial question is one on determinants. Donald Roberston, Thomas McConville, and I resolved some well known questions on formulas determinants involving monomials to powers determine by permutation statistics based on inversions: “Determinental formulas with major indices”, Proceedings of the American Mathematical Society (2021). We propose to generalize these formulas to determinants based on a far-reaching generalization of the inversion statistic in terms of counting vincular and bivincular permutation patterns.
The computational math portion stems from a collaboration between Jonathan Hauenstein and I. Polynomials play an important role in computations throughout mathematics, engineering, and science. The nonlinearity of polynomials is essential in many mathematical models to replicate important behavior while their finiteness properties are crucial to computational mathematics. The overall objective of this research project is to develop new computational approaches related to polynomials for solving fundamental problems involving real aspects of polynomials. Performing real computations are essential to applications but the computational approaches over the real numbers lag far behind the computational approaches over the complex numbers. This research project aims to develop novel approaches that combine efficient numerical algorithms, rigorous mathematical certification, and software to address three fundamental challenges related to polynomial systems.
1. Develop an algorithm based on certified numerical computations to solve the rational non-negativity problem for the cases which are known to be decidable and new cases where decidability is not known.
2. Develop a new computational approach for performing computations with positive dimensional basic semi-algebraic sets via partial differential equations.
3. Develop certified methods for deciding connectivity of points on positive-dimensional basic semi-algebraic sets via certified solving of ordinary differential equations.