We discuss classical and recent research on real and complex coefficient boundary value problems, including some joint work with M. Dindos for so-called p-elliptic divergence form equations

A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions has remained an important open challenge even at the physics level.

I will present a complete solution of this problem in the exponential sense throughout the entire localization regime. Namely, I will describe, with very high precision, the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, a phenomena not even described in the physics literature.

These results lead also to the proof of sharp arithmetic transitions between pure point and singular continuous spectrum, in both frequency and phase, as conjectured since 1994. The talk is based on papers joint with W. Liu

In this talk, we consider non-homogeneous, second order, uniformly elliptic systems of partial differential equations. We show that, within a suitable framework, we can define the fundamental solution and the Green functions on arbitrary open subsets. Moreover, we can prove uniqueness and global estimates that are on par with those of the underlying homogeneous elliptic operator. Our results, in particular, establish the Green functions for Schrodinger, magnetic Schrodinger, and generalized Schrodinger operators with real or complex coefficients on arbitrary domains

The unitary perturbations of a given unitary operator by finite rank d operators can be parametrized by dxd unitary matrices; this generalizes the rank one setting, where the Clark family is parametrized by the scalars on the unit circle. For finite rank perturbations we investigate the functional model of a related class of contractions, as well as a (unitary) Clark operator that realizes such a model representation. We express the adjoint of the Clark operator through a matrix-valued Cauchy integral operator. We determine the matrix-valued characteristic functions of the model (for contractions). In the case of inner characteristic functions results suggest a generalization of the normalized Cauchy transform to the finite rank setting. This presentation is based on joint work with Sergei Treil

Let $m$ be a radial multiplier supported in a compact subset away from the origin. For dimensions $d\ge 2$, it is conjectured that the multiplier operator $T_m$ is bounded on $L^p(R^d)$ if and only if the kernel $K=\hat{m}$ is in $L^p(R^d)$, for the range $1<p<2d/(d+1)$. Note that there are no a priori assumptions on the regularity of the multiplier. This conjecture belongs near the top of the tree of a number of important related conjectures in harmonic analysis, including the Local Smoothing, Bochner-Riesz, Restriction, and Kakeya conjectures. We discuss new progress on this conjecture in dimensions $d=3$ and $d=4$. Our method of proof will rely on a geometric argument involving sizes of multiple intersections of three-dimensional annuli