Georgia Institute of Technology College of Sciences

 We report on recent results regarding the limiting absorption principle for multi-dimensional, massless Dirac-type operators (implying absence of singularly continuous spectrum) and continuity properties of the associated spectral shift function.

We will motivate our interest in this circle of ideas by briefly describing the connection to the notion of the Witten index for a certain class of non-Fredholm operators.

This is based on various joint work with A. Carey, J. Kaad, G. Levitina, R. Nichols, D. Potapov, F. Sukochev, and D. Zanin.

The braid group (which encodes the braiding of n strands) has a canonical projection to the symmetric group (recording where the ends of the strands go). We ask the question: what are the extensions of the symmetric group by abelian groups that arise as quotients of the braid group, by a refinement of this canonical projection? To answer this question, we study a particular twisted coefficient system for the symmetric group, called the integral pair module. In this module, we find the maximal submodule in each commensurability class. We find the cohomology classes characterizing each such extension, and for context, we describe the second cohomology group of the symmetric group with coefficients in the most interesting of these modules. This is joint work with Trevor Nakamura.

Given a discrete set $\Lambda\subseteq\mathbb{R}$ and an interval $I$, define the sequence of complex exponentials in $L^2(I)$, $\mathcal{E}(\Lambda)$, by $\{e^{2\pi i\lambda t}\colon \lambda\in\Lambda\}$.  A fundamental result in harmonic analysis says that if $\mathcal{E}(\frac{1}{b}\mathbb{Z})$ is an orthogonal basis for $L^2(I)$ for any interval $I$ of length $b$.  It is also well-known that there exist sets $\Lambda$, which may be irregular, such that sets $\mathcal{E}(\Lambda)$ form nonorthogonal bases (known as Riesz bases) for $L^2(S)$, for $S\subseteq\mathbb{R}$ not necessarily an interval.

Given $\mathcal{E}(\Lambda)$ that forms a Riesz basis for $L^2[0,1]$ and some 0 < a < 1, Avdonin showed that there exists $\Lambda'\subseteq \Lambda$ such that $\mathcal{E}(\Lambda')$ is a Riesz basis for $L^2[0,a]$ (called basis extraction).  Lyubarskii and Seip showed that this can be done in such a way that $\mathcal{E}(\Lambda \setminus \Lambda')$ is also a Riesz basis for $L^2[a,1]$ (called basis splitting).  The celebrated result of Kozma and Nitzan shows that one can extract a Riesz basis for $L^2(S)$ from $\mathcal{E}(\mathbb{Z})$ where $S$ is a union of disjoint subintervals of $[0,1]$.

In this talk we construct sets $\Lambda_I\subseteq\mathbb{Z}$ such that the $\mathcal{E}(\Lambda_I)$ form Riesz bases for $L^2(I)$ for corresponding intervals $I$, with the added compatibility property that unions of the sets $\Lambda_I$ generate Riesz bases for unions of the corresponding intervals.  The proof of our result uses an interesting assortment of tools from analysis, probability, and number theory.  We will give details of the proof in the talk, together with examples and a discussion of recent developments.  The work discussed is joint with Shauna Revay (GMU and Accenture Federal Services (AFS)), and Goetz Pfander (Catholic University of Eichstaett-Ingolstadt).

The Intermediate Long Wave equation (ILW) describes long internal gravity waves in stratified fluids. As the depth parameter in the equation approaches zero or infinity, the ILW formally approaches the Kortweg-deVries equation (KdV) or the Benjamin-Ono equation (BO), respectively. Kodama, Ablowitz and Satsuma discovered the formal complete integrability of ILW and formulated inverse scattering transform solutions. If made rigorous, the inverse scattering method will provide powerful tools for asymptotic analysis of ILW. In this talk, I will present some recent results on the ILW direct scattering problem. In particular, a Lax pair formulation is clarified, and the spectral theory of the Lax operators can be studied. Existence and uniqueness of scattering states are established for small interaction potential. The scattering matrix can then be constructed from the scattering states. The solution is related to the theory of analytic functions on a strip. This is joint work with Peter Perry.