Mahya Ghandehari, University of Delaware

Abstract: In recent years, certain wavelet-type transformations such as the curvelet or shearlet transformation have gained considerable attention, due to their potential to efficiently handle data with features along edges. In both cases, it was shown that the decay rate of the corresponding transformation coefficients of a tempered distribution identifies the wavefront set of that distribution. Roughly speaking, the wavefront set of a tempered distribution $u$ is the set of points $t\in{\mathbb R}^n$ and directions $\xi\in S^{n-1}$ along which $u$ is not smooth at $t$.

Recently, many efforts have been made aiming to generalize the  characterization of the wavefront set of a tempered distribution, in terms of its continuous wavelet transform, for higher dimensional continuous wavelet transforms. In this talk, we consider the problem of characterizing the Sobolev wavefront set of a distribution constructed using square-integrable representations of ${\mathbb R}^n\rtimes H$ where $H$ can be any suitably chosen dilation group. We tackle two important cases: 1) the mother wavelet is compactly supported, and 2) the mother wavelet has compactly supported Fourier transform.

This talk is based on joint work with Hartmut Fuhr.

Patrick Phelps, Temple University

Abstract: The 2D incompressible inviscid Boussinesq equations model fluid with density variations due to temperature difference. Their similarity to the 3D axisymmetric Euler equations make them a good model for studying the blow up of the  3D Euler equations. Recently, Ignatova published work on the Voigt Regularized 2D Boussinesq equations, and fractional Boussinesq equations which generate statistical solutions to the Boussinesq equations as the regulation parameter tends to zero. We are interested in extending this work to self-similar solutions, and so we rebalance the equations with a time-dependent Voigt regularization. We present results concerning existence, uniqueness, and the structure of self-similar solutions to the 2D Time-dependent Voigt Regularized Bousssinesq equations.

Katrina Morgan, Temple University

Abstract: A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the so-called "string". This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that forward solutions to the wave equation (in an appropriate microlocal sense) do exist. Our techniques involve proving a statement on propagation of singularities and using the resulting estimates to show existence of solutions. This is joint work with Jared Wunsch.