Matrices and tensors that appear in computational mathematics are so often well-approximated by low-rank objects. Since random ("average") matrices are almost surely of full rank, mathematics needs to explain the abundance of low-rank structures. We will give various methodologies that allow one to begin to understand the prevalence of compressible matrices and tensors and we hope to reveal an underlying link between disparate applications. In particular, we will show how one can connect the singular values of a matrix with displacement structure to a rational approximation problem that highlights fundamental connections between polynomial interpolation, Krylov methods, and fast Toeplitz solvers.

Prof. Alex Townsend is an assistant professor at Cornell University in the Mathematics Department. His research is in Applied Mathematics and focuses on spectral methods, low-rank techniques, orthogonal polynomials, and fast transforms. Prior to Cornell, he was an Applied Math instructor at MIT (2014-2016) and a DPhil student at the University of Oxford (2010-2014). He was awarded a SIGEST paper award in 2019, the SIAG/LA Early Career Prize in applicable linear algebra in 2018, and the Leslie Fox Prize in numerical analysis in 2015.