Finite Toeplitz and Hankel matrices are among the most important matrices in mathematics. The asymptotic behavior of the determinants of these matrices can be used to treat a remarkable variety of problems in mathematics, physics, and engineering. Using operator theory and complex analysis, this project aims to describe the asymptotic behavior of these matrices and consider their applications to mathematical physics and random matrix theory, which have connections to emerging quantum technologies. In addition to finite matrices, the project investigates their infinite dimensional counterparts as operators on spaces of functions, and aims to describe their fundamental properties, such as the norm (which measures the size of an operator) and spectrum (which extends the concept of eigenvalues to infinite dimensions). Finally, the project studies the structure of tracial joint spectral measures and uses them to develop a novel approach to resolve Crouzeix's conjecture.

2026

2030