Georgia Institute of Technology College of Sciences

The celebrated Hajnal-Szemerédi theorem gives best possible minimum degree conditions for clique-factors in graphs. There have been some recent variants of this result into several settings, each of which has some sort of randomness come into play. We will give a survey on these problems and the recent developments.

 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.

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).