Abstract: I will review some of the classical results about Toeplitz matrices and determinants (including Szego’s theorems and eigenvalue distributions) and investigate the analogous questions for so-called “multi-Toeplitz” matrices and determinants (which I will define). We will prove the existence of limiting eigenvalue distributions in the multi-Toeplitz setting and pose some open problems.

Abstract: The major problem in the Analysis of PDEs is understanding the nature of singularities of solutions to the PDEs reflecting the natural phenomena. In this talk, I will present new criteria for the removability of the fundamental singularity for the elliptic and parabolic PDEs. The criteria characterize the uniqueness of boundary value problems with singular data, reveal the nature of the harmonic or parabolic measure of the singularity point, asymptotic laws for the conditional Markov processes, and criteria for thinness in minimal- fine topology. The talk will be oriented to a general audience including non-expert faculty and graduate students.

Abstract: Several commonly observed physical and biological systems are arranged in shapes that closely resemble a tessellation of the plane by regular hexagons (honeycomb cluster). Although these shapes are not always the direct product of energy minimization, they can still be understood, at least phenomenologically, as low-energy configurations. In this talk, explicit quantitative estimates on the geometry of every such low-energy planar cluster are provided, showing in particular that the vast majority of the chambers of such clusters must be generalized polygons with six edges, and closely resemble regular hexagons. Part of our argument is a detailed revision of the estimates behind the global isoperimetric principle for honeycomb clusters due to Hales. This talk is based on joint work with M. Caroccia and F. Maggi.