Title: Leibniz-type rules and applications to scattering properties of PDEs.
Abstract: We will discuss Leibniz-type rules in the setting of various function spaces and present applications to scattering properties of certain PDEs. This is joint work with Alex Thomson.
Title: Efficient computation of diffraction near cutoff (Wood anomaly) frequencies.
Abstract: We present an efficient method for computing wave scattering by 2D-periodic diffraction gratings in 3D space near cutoff frequencies, at which a Rayleigh wave is at grazing incidence to the grating. At these frequencies (a.k.a. Wood-anomaly frequencies), the quasi-periodic Green function does not exist, although a solution to the diffraction problem typically does. This is manifest in the divergence of a doubly infinite spatial lattice sum that represents the Green function at generic frequencies. We present a modification of this lattice sum by adding two types of terms to it. The first type adds weighted spatial shifts of the Green function to itself. The shifts are such that the spatial singularities introduced by these terms are located below the grating and therefore out of the spatial domain of interest. With suitable choices of the weights, these terms produce algebraic convergence of the lattice sum. The degree of the algebraic convergence depends on the number of added shifts. The second type of terms are quasi-periodic plane wave solutions of the Helmholtz equation. They reinstate the grazing modes, effectively eliminated by the terms of the first type. These modes are needed for guaranteeing the well-posedness of the boundary-integral equation for the scattered field that involves the Green function. This is joint work with O. Bruno, C. Turc, and S. Venakides.
Title: Dynamics of complex Henon mappings.
Abstract: Complex Henon maps are a special case of polynomial automorphisms of C2 which arise from physical applications and are central objects in the study of holomorphic dynamics in 2D. In this talk we discuss recent progress on Henon maps with a semi-neutral fixed point, which exhibit non-hyperbolic behavior.
Title: Limit of Teichmuller geodesics on Thurston boundary of the Teichmuller space of the unit disk.
Abstract: We continue our study of Thurston boundary of infinite dimensional Teichmuller space by considering the universal Teichmuller space and limits of certain geodesics on the corresponding Thurston boundary. Our result is that any Teichmuller-type geodesic has a unique limit point on Thurston boundary for the weak* topology which is in analogy to the closed surface case where almost every geodesic has a unique limit point(Masur). Surprisingly this convergence fails to be uniform even for some "relatively simple geodesics". Moreover, the limits are not given by the transverse measure of the vertical foliations but rather by the modulus of the vertical foliation. In addition, we consider a non-Teichmuller-type geodesic known as Strebel's chimney. We obtain a unique limit point on Thurston boundary but its support is not contained in the vertical foliation. This is joint work with Hrant Hakobyan.