Analysis Seminar

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.

Irina Mitrea, Temple University

Abstract: In its classical form, the Riemann-Hilbert problem asks for determining two holomorphic functions defined on either side of a surface $\Sigma$, satisfying a boundary condition of transmission type along $\Sigma$ involving a symbol function $\Phi$. In this regard, I will report on recent progress with Marius Mitrea and Michael Taylor describing the Fredholm solvability in the most geometric measure theoretic setting in which such a problem is meaningfully formulated. This involves replacing a complex plane by a Riemannian manifold $\mathcal{M}$, the surface $\Sigma$ by a uniformly rectifiable subset of $\mathcal{M}$, and the Cauchy-Riemann operator by a general Dirac operator on  $\mathcal{M}$ with low regularity assumptions on its coefficients. This topic interfaces with Index Theory on manifolds, and as an application I will discuss the most general Bojarski index formula known to date. 

Artur Andrade, Temple University

Abstract: Elliptic boundary value problems arise naturally in modeling a wide range of physical phenomena, including electrostatics, elasticity, steady-state incompressible fluid flow, and electromagnetism. A powerful tool for the treatment of such problems is the layer potential method, through which matters are reduced to solving a boundary integral equation involving a singular integral operator naturally associated with the domain, and a coefficient tensor for the underlying PDE. When this singular integral operator is compact, the boundary integral equation can be treated using Fredholm Theory.  While systematic progress has been made in the study of second-order elliptic systems along these lines, the case of higher-order elliptic systems remains far less understood.

In this talk, I will present a distinguished coefficient tensor for the polyharmonic operator $\Delta^3$ in all dimensions, and illustrate how the associated singular integral operator is compact on $L^p$ Lebesgue-type spaces, for all integrability exponents $p\in(1,\infty)$, thus opening the door for the employment of Fredholm Theory for the solvability of the Dirichlet Problem for $\Delta^3$ in infinitesimally flat AR domains.

This is an ongoing work with Dorina Mitrea (Baylor University), Irina Mitrea (Temple University), and Marius Mitrea (Baylor University).

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.

Camil Muscalu, Cornell University

Jeongsu Kyeong, Syracuse University

Lancelot Leung, Temple University