Oberseminar Analysis, Mathematische Physik & Dynamische Systeme

Time and Location (if not specified differently)

Tuesday, 14:15-15:45, room M E19
Coffee and tea will be served at 13:45 in room M618.

Talk details

• 13.06.2017 Konstantin Pankrashkin (University Paris-Sud): Self-adjoint operators of the type div sgn grad
Abstract: Being motivated the study of negative-index metamaterials, we will discuss the definition and the spectral properties of the operators given by the differential expressions div h grad in a bounded domain U with a function h which is equal to 1 on a part of U and to a constant b<0 on the rest of U. We will see how the properties of such operators depend on the parameter b and on the geometry of U. In particular, one can have a non-empty essential spectrum. Based on a joint work with Claudio Cacciapuoti and Andrea Posilicano (University of Insubria).

• 27.04.2017 Sjoerd Dirksen (RWTH Aachen): Sparse recovery with heavy-tailed random matrices
Joint with Oberseminar Stochastik, Thursday, 16:00, room M/611.
Abstract: In compressed sensing and high-dimensional statistics, one is faced with the problem of reconstructing a high-dimensional vector x from underdetermined, possibly noisy linear measurements y=Ax. Research from the last decade has shown that this can be done in a computationally efficient way if one knows that the target vector x is sparse or, more generally, comes from a "low-complexity" model. The best known reconstruction results are known for ?well-behaved? random measurement matrices, e.g., Gaussian matrices.
In this talk I will consider the problem of recovering x via a convex program, called \ell_p-constrained basis pursuit, in the scenario that A contains heavy-tailed random variables. I will present recent work that shows that under surprisingly light conditions on the distribution on the entries, one can reconstruct x from an optimal number of measurements. If time permits, I will show an application to reconstruction from quantized heavy-tailed measurements.
I will not assume any prior knowledge of compressed sensing during the presentation.
Based on joint work with Guillaume Lecué (Ecole Polytechnique) and Holger Rauhut (RWTH Aachen University).

• 14.02.2017 Christopher Classen (Dortmund): Subnormale Lösungen der vierten Painlevéschen Differentialgleichung
Zusammenfassung: Die Lösungen der vierten Painlevéschen Differentialgleichung sind entweder rationale Funktionen oder in der komplexen Ebene transzendente meromorphe Funktionen endlicher Ordnung. Betrachtet werden die Lösungen deren Zählfunktion n(r,w)=O(r^2) genügt, die sogenannten subnormalen Lösungen. Mit Hilfe der Hermite-Weber Differentialgleichung lassen sich unter dem Begriff Hermite-Weber Lösung alle Lösungen zusammenfassen, die sich aus Lösungen der Hermite-Weber Differentialgleichung unter sukzessiver Anwendung von Bäcklundtransformationen ergeben. Es gelingt die Zählfunktion signifikant zu reduzieren, so dass man nach endlich vielen Anwendungen geeigneter Bäcklundtransformationen in einer Hermite-Weber Differentialgleichung landet. Da dies für alle subnormalen Lösungen gelingt, folgt als Hauptresultat, dass jede subnormale Lösung der vierten Painlevéschen Differentialgleichung eine Hermite-Weber Lösung ist.

• 07.02.2017 Martin Vogel (Paris) Spectral statistics of non-normal operators subject to small random perturbations
Abstract: It is well known that the spectrum non-normal operators can be highly unstable even under tiny perturbations. Exploiting this phenomenon it was shown in recent works by Hager, Bordeaux-Montrieux, Sjöstrand, Christiansen and Zworski that one obtains a probabilistic Weyl law for a large class of non-normal semiclassical pseudo-differential operators after adding a small random perturbation. We will discuss some recent results obtained in collaboration with Stéphane Nonnenmacher concerning the local statistics of eigenvalues of such operators. That is the statistical interaction between the eigenvalues on the scale of their average spacing.

• 31.01.2017 Christian Jäh (Loughborough University): Hyperbolic systems with variable multiplicities
Abstract

• 17.01.2016 Albrecht Seelmann (Mainz): Invariant graph subspaces and block diagonalization
Abstract: The problem of decomposition for unbounded self-adjoint 2x2 block operator matrices by a pair of orthogonal graph subspaces is discussed. As a byproduct of our consideration, a new block diagonalization procedure is suggested that resolves related domain issues. The results are discussed in the context of a two-dimensional Dirac-like Hamiltonian.

• 10.01.2017 Mostafa Sabri (Strasbourg): Quantum ergodicity for the Anderson model on regular graphs
Abstract: In this talk I will discuss a result of delocalization for the Anderson model on the regular tree (Bethe lattice). The Anderson model is a random Schrodinger operator, where we add a random i.i.d. perturbation to the adjacency matrix. Localization at high disorder is well understood today for a wide variety of models, both in the sense of a.s. pure point spectrum with exponentially decaying eigenfunctions, and in a dynamical sense. Delocalization remains a great challenge. For tree models, it is known that for weak disorder, large parts of the spectrum are a.s. purely absolutely continuous, and the dynamical transport is ballistic. In this work, we try to complete the picture by proving that in such AC regime, the eigenfunctions are also delocalized in space, in the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed (as opposed to the exponential decay in the localization regime). The precise result is a quantum ergodicity theorem. A different criterion was obtained by Geisinger. (This is a joint work with Nalini Anantharaman).

• 06.11.2016: Matthias Täufer (TU Dortmund), Uniform Sobolev Estimates
Abstract: This talk follows an article by Kenig, Ruiz and Sogge (Duke Math. J., 1987) in which "Uniform Sobolev estimates" for constant coefficient second order partial differential operators, i.e. estimates with a constant that does not depend on the lower order terms, are proven. They imply Carleman estimates which themselves lead to unique continuation of eigenfunctions. The main difficulty is to deal with the lower order terms. This is accomplished by a localization in Fourier space and tools from harmonic analysis such as the Littlewood-Paley decomposition and Fourier restriction Lemmas.
• 29.11.2016: Christoph Schumacher (TU Dortmund), Bedingt unabhängige Kopien von Zufallsvariablen
Abstract: In vielen Fällen braucht man unabhängige Kopien einer Zufallsvariable. Diese sind leicht mit Hilfe deines Produktmaßes konstruiert. In diesem Vortrag zeige ich, wie man mit Hilfe bedingter Wahrscheinlichkeiten Kopien einer Zufallsvariable anlegen kann, deren Werte teils vom Original kopiert und teils so unabhängig wie möglich ergänzt werden. Obwohl das Lemma ein Bestandteil des Beweises zur uniformen Existenz der integrierten Zustandsdichte zufälliger Schödingeroperatoren auf endlich erzeugten amenablen Gruppen ist, lässt es sich geschlossen formulieren und beweisen.
• 08.11.1016 & 15.11.2016: Michela Egidi (TU Dortmund), Quantitative uncertainty principles on multidimensional tori: Logvinenko-Sereda type theorems.
Abstract: In this talk we present unique continuation (or uncertainty) principles for the class of $L^p$ functions on multidimensional tori with Fourier transform supported on a (finite) union of parallelepipeds, exhibiting the explicit dependence on the model parameters. These results are inspired by the Logvinenko-Sereda Theorem, then generalized by Kovrijkine. The proofs rely on method of Fourier analysis and complex function theory.