# Oberseminar Analysis, Mathematische Physik & Dynamische Systeme

### Time and Location (if not specified differently)

Tuesday, 14:15-15:15, room M/611
Coffee and tea will be served at 13:30 in room M/618.

### Talk details

• 10.04.2018 (14:15) Norbert Peyerimhoff (Durham University): Spectra of signed Laplacians on graphs
Abstract: In this talk I will give a survey on spectral aspects of graphs with signatures. Signatures are an extra structure on the set of oriented edges which give rise to particular lifts and signed Laplacians. We will also talk about a generalisation of Cheeger constants to signed (or magnetic) Cheeger constants and will discuss connections to the spectrum. The presented material is chosen from joint work with Carsten Lange, Shiping Liu, Olaf Post, and Alina Vdovina.

• 03.04.2018 (14:15) Christoph Richard (FAU Erlangen Nürnberg): Fourier analysis of unbounded Radon measures and diffraction theory
Abstract: Fourier analysis of unbounded Radon measures on locally compact Abelian groups was developed by Argabright and de Lamadrid in the 70's. We review part of that theory, with focus on its relation to Fourier analysis of tempered distributions in Euclidean space. We also discuss its application to diffraction of model set Dirac combs and more general measures. This is based on joint work with Nicolae Strungaru (Edmonton).
Vortrag im Rahmen des Graduiertenkollegs 2131 'Phänomene hoher Dimensionen in der Stochastik - Fluktuationen und Diskontinuität'

• 23.01.2018 Yulia Petrova (St. Petersburg): Exact $$L_2$$-small ball probabilities for finite-dimensional perturbations of Gaussian processes
Abstract: I consider the problem of small ball probabilities for Gaussian processes in $$L_2$$-norm. I focus on the processes which are important in statistics (e.g. Kac-Kiefer-Wolfowitz processes), which are finite dimentional perturbations of Gaussian processes.
Depending on the properties of the kernel and perturbation matrix I consider two cases: non-critical and critical.

• For non-critical case I prove the general theorem for precise asymptotics of small deviations.
• For a huge class of critical processes I prove a general theorem in the same spirit as for non-critical processes, but technically much more difficult.
• At the same time a lot of processes naturally appearing in statistics (e.g. Durbin, detrended processes) are not covered by those general theorems, so I treat them separately using methods of spectral theory and complex analysis.

Vortrag im Rahmen des Graduiertenkollegs 2131 'Phänomene hoher Dimensionen in der Stochastik - Fluktuationen und Diskontinuität'

• 17.01.2018 (11:00 (sharp), room M1011) Max Kämper (TU Dortmund): Einflüsse veränderter Energie-Impuls-Beziehungen auf Neutrinooszillationen
Abstract: Die relativistische Energie-Impuls-Beziehung $$E^2=m^2c^4+p^2c^2$$ ist bei den bisher gemessenen Energien und Impulsen gut überprüft, doch bei hohen Impulsen könnte sie durch Strukturen im Bereich der Plancklänge verändert werden, eine Gitterstruktur könnte beispielsweise zu einer periodischen Beziehung führen. Die direkte Messung von Energie-Impuls-Beziehungen ist bei großen Impulsen schwer durchzuführen, aber mithilfe von Neutrinooszillationen können sie indirekt gemessen werden. Durch den Ebene-Wellen- sowie den Wellenpaket-Formalismus können die Auswirkungen von geänderten E-p-Beziehungen berechnet und Kohärenzbedingungen aufgestellt werden. Als Resultat kann die Existenz von einzelnen Masseneigenzuständen mit geänderten E-p-Beziehungen in manchen Fällen sicher, in manchen Fällen nahezu ausgeschlossen werden. Es kann nicht ausgeschlossen werden, dass alle Neutrinos eine periodische Beziehung besitzen, es können jedoch obere Grenzen für die Oszillationslänge berechnet werden.
Vortrag im Rahmen des Graduiertenkollegs 2131 'Phänomene hoher Dimensionen in der Stochastik - Fluktuationen und Diskontinuität'

• 16.01.2018 Hafida Laarsi (Hagen): On $$L^p$$-maximal regularity for non-autonomous evolution equations
Abstract: We consider non-autonomous parabolic equation of the form $\dot u(t)+A(t)u(t)=0, \ t\in[0,T],\ u(0)=u_0.$ Here $$A(t), t\in [0,T]$$, are associated with a non-autonomous sesquilinear form $$a(t,\cdot,\cdot)$$ on a Hilbert space $$H$$ with constant domain $$V\subset H$$. We give a brief introduction to $$L^p$$-maximal regularity for non-autonomous linear evolution equations of the form. Furthermore, we study some fundamental theoretical properties of the associated evolution family. Recall that it is well known that, under suitable conditions, the solution of a non-autonomous linear evolution equation may be given by a strongly continuous evolution family. The later is in fact the non-autonomous counterpart of operator semigroup in the well-posedness theory of non-autonomous evolution equations.
Vortrag im Rahmen des Graduiertenkollegs 2131 'Phänomene hoher Dimensionen in der Stochastik - Fluktuationen und Diskontinuität'

• 12.1.2018 Christoph Schumacher (TU Dortmund): The asymptotic behavior of the ground state energy of the Anderson model on large regular trees
Abstract: The Anderson model was invented 1985 by Anderson and describes the quantum mechanical motion of a particle in a random potential in $$Z^d$$. It is related to random walks in a random environment. The Anderson model on regular trees was introduced 1973 by Abou-Chacra, Thouless and Anderson. We give a detailed description of the ground state energy on large finite symmetric subtrees. This is joint work with Francisco Hoecker-Escuti (TU Hamburg-Harburg).

• 9.1.2018 Peter Mühlbacher (IST Austria): Bounds on the Norm of Wigner-type Random Matrices
Abstract: We consider a Wigner-type ensemble, i.e. large hermitian $$N\times N$$ random matrices $$H=H^*$$ with centered independent entries and with a general matrix of variances $$s_{xy}=\mathbb E|H_{xy}|^2$$. The norm of $$H$$ is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of $$S$$ that substantially improves the earlier bound $$2\| S\|^{1/2}_\infty$$ given in \cite{ELK16}. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.
Vortrag im Rahmen des Graduiertenkollegs 2131 'Phänomene hoher Dimensionen in der Stochastik - Fluktuationen und Diskontinuität'

• 19.12.2017 Julian Großmann (TU Hamburg-Harburg): Oscillation theory for Jacobi operators with applications to high dimensional random operators
Abstract: Sturm-Liouville oscillation theory is studied for Jacobi operators with block entries given by elements of a unital $$C^*$$-algebra. New results in this general framework are developed and eventually applied to certain high-dimensional random Schrödinger operators. It is shown that the integrated density of states of the Jacobi operator is approximated by a certain generalised winding number. These results are based on arXiv:1706.07498.

• 12.12.2017 Mira Shamis (Queen Mary University of London): On the Wegner orbital model
Abstract: The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential is Gaussian, and prove three results: localisation at strong disorder, a Wegner-type estimate on the mean density of eigenvalues, and a Minami-type estimate on the probability of having multiple eigenvalues in a short interval. The last two results are proved in the more general setting of deformed block-Gaussian matrices, which includes a class of Gaussian band matrices as a special case. Emphasis is placed on the dependence of the bounds on the number of orbitals. As an additional application, we improve the upper bound on the localisation length for one-dimensional Gaussian band matrices. https://arxiv.org/abs/1608.02922v1
Guest of the Research Training Group (RTG) High-dimensional Phenomena in Probability.

• 5.12.2017 Georgi Raikov (Pontificia Universidad Católica de Chile): Lifshits tails for randomly twisted quantum waveguides
Abstract: I will consider the Dirichlet Laplacian on a three-dimensional twisted waveguide with random Anderson-type twisting. I will discuss the Lifshits tails for the related integrated density of states (IDS), i.e. the asymptotics of the IDS as the energy approaches from above the infimum of its support. In particular, I will specify the dependence of the Lifshits exponent on the decay rate of the single-site twisting.
The talk is based on joint works with Werner Kirsch (Hagen) and David Krejcirik (Prague). The partial support of the Chilean Science Foundation Fondecyt under Grant 1170816 is gratefully acknowledged.

Guest of the Research Training Group (RTG) High-dimensional Phenomena in Probability.

• 1.12.2017 Martin Tautenhahn (Friedrich-Schiller-Universität Jena und Technische Universität Chemnitz): Anderson models and generalization to locally finite graphs and correlated potentials
time: 13:00 - 14:00, room: M511
Abstract: The modelling of disordered solids in condensed matter physics leads to the study of random Schrödinger operators. The prototype of such an operator is the Anderson model which describes a spinless electron moving in a static random electric potential $$V_\omega$$ on the state space $$\ell^2 (\mathbb{Z}^d)$$. The potential values are assumed to be independent identically distributed random variables. With this simple model it is possible to describe the transition from metal to insulator under the presence of disorder. In a first part of the talk, we introduce the Anderson model, discuss basic properties thereof, and review classical results on localization via the so-called fractional moment method and the multiscale analysis.
The second and the third part of the talk are devoted to generalizations of the classical Anderson model. First, we discuss a generalization to locally finite graphs $$G = (V,E)$$ instead of $$\mathbb{Z}^d$$. We will elaborate geometric conditions on the graph $$G$$, such that localization still holds in the case of sufficiently large disorder. Second, we discuss a generalization of the Anderson model to the case where the potential values at different lattice sites are correlated random variables, in particular, where $$V_{\cdot} : \Omega \times \mathbb{Z}^d \to \mathbb{R}$$ is a Gaussian process with sign-indefinite covariance function.

Guest of the Research Training Group (RTG) High-dimensional Phenomena in Probability.

• 19.09.2017 Daniel Schindler: Diffusion Maps: From Classification to Molecular Dynamics

• 17.08.2017 Mike Parucha (Bonn): Characterization of the Ricci flow
(time: 14:30, room number M611)
Abstract

• 17.08.2017 Christian Horvat: Stochastische Modelle in der Populationsgenetik
(time: 13:00, room M611; joint with Graduiertenkolleg 2131: Phänomene hoher Dimensionen in der Stochastik - Fluktuationen und Diskontinuität)
Abstract: Was sind die natürlichen Mechanismen, die verantwortlich sind, dass sich Gene verändern oder nicht? Im Zusammenspiel mit der wohlbekannten natürlichen Selektion Darwins, beeinflussen Migration und Mutation die Dynamik von Erbgutinformationen. Weniger bekannt hingegen ist der Zufall an sich: er hat besonders grossen Einfluss auf die genetische Vielfalt bei kleinen Populationsgrössen. Ich werde Ihnen die wichtigsten Populationsmodelle vorstellen, die versuchen diesen Mechanismus des Zufalls mathematisch zu beschreiben und zu verstehen: das Wright-Fisher Modell, das Moran-Modell, der lookdown Prozess und deren gemeinsamer Grenzwertprozess: die Wright-Fisher Diffusion. Zum Schluss werde ich kurz einen weiteren Mechanismuns vorstellen, der erst kürzlich entdeckt wurde und gegen den Verlust von genetischer Vielfalt ankämpft, also gegen den Einfluss des Zufalls: der sog. seed-bank Effekt.

• 18.07.2017 Christoph Schumacher (Dortmund): Concentration of Measures in Random Schrödinger Operators

• 11.07.2017 Hendrik Vogt (Hannover): $$L_\infty$$-Absch"atzungen für die Torsionsfunktion
Abstract

• 25.05.2017 Frederik Drewin (Wuppertal): Funnel-Steuerung für randgesteuerte Wärmeleitungsgleichungen
Wir betrachten ein System, welches sich durch die Wärmeleitungsgleichung mit Neumann-Randbedingung auf einem glatten Gebiet im $$R^d$$ beschreiben lässt. Wie wir im Vortrag sehen werden, lässt sich darauf die sogenannte *Funnel(zu deutsch: Trichter)-Steuerung anwenden. Dadurch erhält man, dass die Abweichung zwischen einem vorgegebenen Referenz-Signal und dem tatsächlich vorliegenden Signal im Funnel liegt. Folglich strebt diese Abweichung also asymptotisch gegen 0. Dazu wird im Verlauf des Vortrags ebenfalls gezeigt, dass sich die Wärmeleitungsgleichung als ein sogenanntes Randsteuerungssystem und auch als ein wohlgestelltes System auffassen lässt.

• 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 $$2\times2$$ 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.

### Kontakt

TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund

Sie finden uns auf dem sechsten Stock des Mathetowers .

Sekretariat: Janine Textor
Raum M 620

E-mail:
janine.textor@tu-dortmund.de

Tel. (0231) 755-3063
Fax (0231) 755-5219