# Oberseminar Analysis, Mathematische Physik & Dynamische Systeme

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

Tuesday, 14:15-16:15, online via Zoom

### Talk details

• 18.05.2021, 14:00-14:45, online via Zoom, Wolfgang Spitzer (FernUni Hagen): Entanglement entropy of the ideal Fermi gas in a magnetic field I
Afterwards 15 min discussion

18.05.2021, 15:00-15:45, online via Zoom, Paul Pfeiffer (FernUni Hagen): Entanglement entropy of the ideal Fermi gas in a magnetic field II
Afterwards 15 min discussion

• 04.05.2021, 14:00-14:45, online via Zoom, Anton Klimovsky (Uni Duisburg-Essen): Complex Random Energy Models
Abstract: Random energy models (REM) suggested by B. Derrida turned out to be instrumental in the rigorous progress on statistical physics of disordered systems. Building on their success and motivated by interference phenomena, quantum physics, the Lee-Yang theory of phase transitions, the conjectured links with the Riemann zeta function and random matrix theory, we study several complex-valued versions of the REM. Specifically, we focus on the REM, the so-called generalized REM and on the so-called branching Brownian motion energy model. These are energy-based models of disordered systems, where the energy function is a complex-valued Gaussian process on a tree. In the two latter models, the energy function has strong correlations, while in the REM, the energy function is white noise. This allows us to explore the universality classes of the models by varying the strength of correlations. In all models, we identify the phase diagram, the fluctuations and the distribution of the zeros of the random partition function. To this end, we prove a range of non-standard limit theorems for the sums of strongly correlated random variables. Time permitting, we will discuss an ongoing work on models with microscopic interactions.
The talk is based on joint works with Zakhar Kabluchko and Lisa Hartung.

Afterwards 15 min discussion

04.05.2021, 15:00-15:45, online via Zoom, Joachim Kerner (FernUni Hagen): Remarks on the asymptotic behaviour of the spectral gap of one-dimensional Schrodinger operators
Abstract: In this talk we discuss recent results on the asymptotic behaviour of the spectral gap of one-dimensional Schrodinger operators in the limit of large intervals. The spectral gap, being defined as the difference between the first two eigenvalues, is a classical object in the spectral theory of operators but it also appears frequently in more applied settings. As a main objective, we will derive upper and lower bounds on the spectral gap for certain (and quite general) classes of potentials. Doing this, we will also come across some relatively surprising findings (partially, this is based on joint work with M. Täufer).
Afterwards 15 min discussion

• 20.04.2021, 14:00-14:45, online via Zoom, Christian Seifert (TU Hamburg): Stabilisierbarkeit und schwache Beobachtbarkeit in Banachräumen
Abstract: Wir betrachten Systeme $x'(t) = Ax(t) + Bu(t)$, $x(0) = x_0$ mit einem Erzeuger $A$ einer $C_0$-Halbgruppe auf einem Banachraum $X$ und $B\in \mathcal{L}(U,X)$, wobei $U$ ebenfalls ein Banachraum ist. Wir studieren zum einen das Konzept Stabilisierbarkeit, d.h. Existenz eines $K\in \mathcal{L}(X,U)$, so dass $A+BK$ eine exponentiell stabile Halbgruppe erzeugt; anders formuliert, die Kontrolle $u(t) = Kx(t)$ stabilisiert das System. Zum anderen betrachten wir schwache Beobachtbarkeit, d.h. ist $Y$ ein weiterer Banachraum, $C\in\mathcal{L}(X,Y)$, und $y(t) = Cx(t)$, dann soll $\|x(t)\|$ kontrollierbar durch eine $L_r$-Norm von $y$ und durch die Norm von $x_0$ sein. Wir stellen die Beziehung zwischen den beiden Konzepten und sowie hinreichende Kriterien für sie dar.

Afterwards 15 min discussion

20.04.2021, 15:00-15:45, online via Zoom, Greta Marino (TU Chemnitz): Existence and multiplicity results for double phase problems with nonlinear boundary conditions
Abstract: In this talk we consider double phase problems with nonlinear boundary conditions possibly exhibiting a gradient dependence. Under quite general assumptions we prove existence results for such problems, where the perturbations satisfy a suitable behavior in the origin and at infinity. In the second part we focus on a class of Kirchhoff type problems set on a double phase framework, for which the existence of infinitely many solutions is ensured.
This talk is based on joint works with S. El Manouni, A. Fiscella, A. Pinamonti and P. Winkert.

Afterwards 15 min discussion

• 17.03.2020, 14:15, M/511, Postponed due to Corona Ivan Moyano (Université de Nice Sophia-Antipolis): Propagation of smallness and control for heat equations
Abstract: In this note we investigate propagation of smallness properties for solutionsto heat equations. We consider spectral projector estimates for the Laplaceoperator with Dirichlet or Neumann boundary conditions on a Riemanian manifoldwith or without boundary. We show that using the new approach for thepropagation of smallness from Logunov-Malinnikova [7, 6, 8] allows to extendthe spectral projector type estimates from Jerison-Lebeau [3] from localisationon open set to localisation on arbitrary sets of non zero Lebesgue measure; wecan actually go beyond and consider sets of non vanishing d -- δ(δ > 0 small enough) Hausdorff measure. We show that these new spectralprojector estimates allow to extend the Logunov-Malinnikova's propagation ofsmallness results to solutions to heat equations. Finally we apply theseresults to the null controlability of heat equations with controls localised onsets of positive Lebesgue measure. A main novelty here with respect to previousresults is that we can drop the constant coefficient assumptions (see [1, 2])of the Laplace operator (or analyticity assumption, see [4]) and deal withLipschitz coefficients. Another important novelty is that we get the first (nonone dimensional) exact controlability results with controls supported on zeromeasure sets. (Joint work with N. Burq)

• 28.01.2020, 14:15, M/511, Sebastian Liedtke (TU Dortmund): Minkowski-Billards

• 21.01.2020, 14:15, M/511, Baris Evren Ugurcan (Bergische Universität Wuppertal): Renormalization of the Anderson Hamiltonian and associated stochastic PDE

• 07.01.2020, 14:45, Arianna Giunti (Universität Bonn): Homogenization in randomly perforated domains
Abstract

Kaffee und Tee in Raum M/618 heute von 14 Uhr bis zum Vortrag.

• 17.12.2019, 14:15, M/511, Christian Rose (Max Planck Institute for Mathematics in the Sciences): Unique continuation principles on Riemannian manifolds
Abstrakt: Unique continuation principles constitute a very active field in control theory or the theory of random Schrödinger operators. Usually, such ucp are proved by Carleman estimates applied to generalized eigenfunctions. Carleman estimates usually depend on ellipticity and Lipschitz assumptions on the symbol of the partial differential operator under consideration. In the case of Riemannian manifolds there exist Carleman estimates and ucp for the Laplace-Beltrami operator similar to elliptic operators in $$\mathbb{R}^d$$. Those depend of course on elliptic and Lipschitz assumptions on the given Riemannian tensor. This circumstance makes it impossible to derive ucp depending on curvature restrictions, since it is not known how, e.g., Ricci curvature restrictions translate into uniform assumptions for the metric. I will present ucp for non-negatively Ricci curved manifolds as well as compact manifolds with Ricci curvature bounded below for small energies without using Carleman. This is joint work in progress with Martin Tautenhahn.

• 08.10.2019 & 15.10.2019, 14:15, M/511, Alexander Dicke (TU Dortmund): Unique Continuation with Weak Type Lower Order Terms
Abstrakt: In this talk we sketch the proof of the main result of Lu and Wolff (Potential Anal., 1997). Roughly speaking, it is shown that a function $$u\in W^{2,p}_{\mathrm{loc}}$$ vanishes identically in some domain if it vanishes on an open subset and there a weights $$A$$ and $$B$$ that lie in some appropriate weak Lebesgue-space such that $$|\Delta u|\leq A|u|+B|\nabla u|$$ holds true.

• 20.08.2019, 14:15, M/611, Matthias Täufer (Queen Mary, University of London): Multi scale analysis in the weak coupling regime.

• In cooperation with the Research Training Group 2131:
08.07.2019, 17:00, M/611, David Damanik (Houston): The Fürstenberg-Ishii Criterion for a Positive Lyapunov Exponent and Applications to Anderson Localization
Abstrakt: We discuss a criterion for random products of SL(2,R) matrices to have a positive Lyapunov exponent that is inspired by work of Ishii and that provides a sufficient condition for an application of the well-known Fürstenberg theorem. Applications of this criterion include localization results for the continuum 1D Anderson model, random operators on random radial trees, and Schrödinger operators with general random local point interactions.

• 28.06.2019, 14:15, M/511, Illia M. Karabash (NAS of Ukraine): Hamilton-Jacobi and Euler-Lagrange equations arising in optimization of resonances.
Abstrakt: Optical resonators with small decay rate (or high-Q cavities) are important components in the contemporary optical engineering because they enhance intrinsically small light-matter interactions. The related mathematical problem is to design a photonic crystal structure that, under certain constraints, generates a resonance as close as possible to the real line. While the engineering and computational aspects of the problem have drawn great attention since first fabrication successes more than a decade ago, the analytic background of spectral optimization for non-Hermitian eigenproblems is still in the stage of development. It is planned to present the recently developed theory of Pareto optimizers and associated Euler-Lagrange equations. In the case of layered optical cavities, we will demonstrate that the resonance optimization can be partially reduced to minimum time control and derive the corresponding Hamilton-Jacobi-Bellman equation. On the other hand, Maximum Principle can be combined with shooting method to compute effectively optimal symmetric structures. We also discuss briefly the first steps in the optimization of multidimensional resonances. The talk is partially based on the joint works (Albeverio, Karabash, 2017), (Karabash, Logachova, Verbytskyi, 2017), (Karabash, Koch, Verbytskyi, 2018), and (Eller, Karabash, 2019).

• 21.06.2019, 14:15, M/511, Merdan Artykov (TU Dortmund): Limit theorems for random walks on non-compact Grassmannians with growing dimensions

• 18.06.2019, 14:15, M/511, Werner Kirsch (FU Hagen): The Curie-Weiss model - a simple model for magnetism
Abstrakt: The Curie-Weiss model is probably the easiest model for magnetism, it shows a phase transition between a non-magnetic and a magnetic phase, yet it can be solved rather explicitly. In this talk we introduce this model and show some of its most important properties using elementary methods. At the end of the talk we present two recent developments using the Curie-Weiss model, one on voting theory and one on random matrices.

• 14.06.2019, 14:15, M/511, Christoph Schumacher (TU Dortmund): Zum Grundzustand des Anderson-Operators auf dem Bethe-Gitter
Basierend auf einer gemeinsamen Arbeit mit Francisco Hoecker-Escuti.
Abstrakt: Der Laplace-Operator auf dem regulären Baum mit Knotengrad $$k+1$$, auch Bethe-Gitter genannt, hat die Grundzustandsenergie $$(\sqrt{k}-1)^2$$. In diesem Vortrag möchte ich ein Cut-Off-Phänomen für die Grundzustandsenergie des Anderson-Operators, also die Summe aus dem Laplace-Operator und einem zufälligen Potential, auf dem Bethe-Gitter vorstellen. Diese stochastische Beschreibung der Grundzustandsenergie kann zum Beweis von Lifshitz-Verhalten der IDS des Anderson-Operators auf dem Bethe-Gitter genutzt werden.

• 28.05.2019, 14:15, M/511, Christoph Schumacher (TU Dortmund): Die Pastur-Shubin-Formel auf Graphen
Abstrakt: Stattet man die Menge der lokal endlichen, dekorierten Graphen mit Wurzel mit der sogenannten lokalen Topologie aus, so erhält man einen polnischen Raum. Nach einer kurzen Einführung dieses Raumes nutzen wir ihn zur Definition zufälliger Graphen, definieren auf diesen selbstadjungierte Operatoren und studieren deren integrierte Zustandsdichte (IDS) in Abhängigkeit vom zugrundeliegenden Graphen. Ziel des Vortrags ist die Stetigkeit der IDS bezüglich der lokalen Topologie.

• 21.05.2019, 14:15, M/511, Alexander Dicke (TU Dortmund): Zufällige Divergenz-Typ Operatoren
Abstrakt: Wir betrachten zufällige Operatoren der Form $$H_{\omega} = -\mathrm{div}(1+V_{\omega})\mathrm{Id}\nabla$$. Dabei ist $$V_{\omega}$$ ein geeignet gewähltes, nicht-negatives, zufälliges Potential mit kleinem Träger.

• 14.05.2019, 14:15, M/511, Albrecht Seelmann (TU Dortmund): Lokalisierung an Bandkanten für nicht-ergodische zufällige Schrödingeroperatoren

• 30.04.2019, 14:15, M/511, Max Kämper (TU Dortmund): Von den Gesetzen der großen Zahlen zu Glivenko-Cantelli
Abstrakt: Herr Max Kämper trägt über seine Bachelorarbeit vor. Sie beschäftigt sich mit der Konvergenz von Zufallsereignissen anhand der Gesetze der großen Zahlen, des Birkhoff'schen Ergodensatz und des Satz von Glivenko-Cantelli.

• 23.04.2019, 14:15, M/511, Hong Van Le (Prag): Novikov homology and Novikov fundamental group
Abstrakt: In my talk I shall first give a short survey on Novikov homology and its applications. Then I shall outline the construction of Novikov fundamental group, which is a refinement of Novikov homology, and its applications that have been introduced and investigated in our recent joint work with Jean Francois Barraud, Agnes Gabled and Roman Golovko https://arxiv.org/abs/1710.10353.

• 09.04.2019, 14:15, M/511, Leonid Zeldin (TU Dortmund): Danzig-Selektor
Abstrakt: Herr Leonid Zeldin trägt über seine Masterarbeit vor. Diese beruht auf einer Arbeit von Candes und Tao, bei der Ideen aus dem Compressive Sensing in der Statistik angewendet werden.

• 04.03.2019, 11:15, M/511, Stephan Schmitz (Universität Koblenz-Landau, Campus Landau): Indefinite Quadratische Formen und das Tan-2-Theta-Theorem der Fluid Mechanik
Abstrakt: Im ersten Teil des Vortrags wird die Korrespondenz zwischen Operatoren und quadratischen Formen beleuchtet. Der Darstellungssatz von Riesz garantiert eine eins-zu-eins Korrespondenz zwischen beschränkten selbstadjungierten Operatoren und symmetrischen Formen. Für indefinite unbeschränkte Formen/Operatoren ist diese Situation komplizierter.
Im zweiten Teil wird Ladyzhenskays's bemerkenswerter Stabilitäts-Satz der Fluidmechanik präsentiert: Für kleine Reynoldszahlen werden Strömungen im Grenzwert stationär.
Als mögliche Erklärung dieser Stabilität wird gezeigt, dass die Rotation spektraler Teilräume des Stokes-Operators durch die Reynoldszahl beschränkt ist, das Tan-2-Theta-Theorem der Fluid Mechanik.
Der Vortrag basiert auf gemeinsamer Arbeit mit L. Grubišić, V. Kostrykin, K. A. Makarov und K. Veselić.

• 26.02.2019, 14:15, Fabian Schwarzenberger (HTW Dresden): Tiling Theorems and Applications to the Integrated Density of States

• 26.02.2019, 15:00 Martin Tautenhahn (TU Chemnitz): A sufficient condition for observability in Banach spaces and application to $$L^p$$-spaces

• 15.01.2019 Nicole Kersten (Mastervortrag): Existenz wandernder Gebiete in komplexen dynamischen Systemen

• 08.01.2019 Iván Moyano (Cambridge): Spectral inequalities

• 18.12.2018 Lars Schroeder (TU Dortmund): Eine polynomielle Efron-Stein-Ungleichung für Zufallsmatrizen
Abstrakt: Wir leiten eine polynomielle Schranke für den Erwartungswert der Norm einer Zufallsmatrix her. Dabei nutzen wir die Methode der austauschbaren Paare und der Stein-Paare und erhalten eine obere Schranke durch eine bedingte Varianz, die an die Efron-Stein-Ungleichung erinnert.

• 04.12.2018 Jean Gutt (Köln): Knotted symplectic embeddings
Abstract: I shall discuss a joint result with Mike Usher, showing that many toric domains X in the 4-dimensional euclidean space admit symplectic embeddings f into dilates of themselves which are knotted (i.e. non-equivalent to the inclusion) in the strong sense that there is no symplectomorphism of the target that takes f(X) to X.

• 27.11.2018 Christian Seifert (TU Hamburg): Observability for Systems in Banach spaces
Abstract: In this talk we study sufficient conditions for obserability of systems in Banach spaces. In an abstract Banach space setting we show that an uncertainty relation together with a dissipation estimate implies an observability estimate with explicit dependence on the model parameters. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider elliptic operators on $$L_p(\mathbb{R}^d)$$ and on $$L_p(\mathbb{T}^d)$$ for $$p \in (1, \infty)$$. Combined with the well-known relation between observability and controllability we derive sufficient conditions for null-controllability and bounds on the control cost. The talk is based on joint work with Dennsi Gallaun (TU Hamburg) and Martin Tautenhahn (TU Chemnitz).

• 20.11.2018 Albrecht Seelmann (TU Dortmund): Approximation durch Ausschöpfungen für das Kontrollproblem der Wärmeleitungsgleichung auf unbeschränkten Gebieten
Abstrakt: Wir betrachten das Null-Kontrollproblem für die Wärmeleitungsgleichung auf beschränkten und unbeschränkten Gebieten. Wir zeigen, dass für eine schwach konvergente Folge von einer Ausschöpfung zugeordneten Null-Kontrollfunktionen die Grenzfunktion eine Null-Kontrolle auf dem Grenzgebiet darstellt. Dies erlaubt es, das Kontrollproblem auf unbeschränkten Gebieten auf das Problem auf einer Folge von beschränkten Gebieten zu reduzieren. Der Vortrag basiert auf einer gemeinsamen Arbeit mit I. Veselić.

• 13.11.2018 Michela Egidi (TU Dortmund): Geometric conditions for the null-controllability of Ornstein-Uhlenbeck Equations on $$\mathbb{R}^d$$

• 06.11.2018, 14:00, Dennis Malcherczyk (TU Dortmund): Asymptotische Verteilung von vollen Datentiefen
Abstract

• 05.11.2018 (Monday, time: 17:00, location: room WSC-S-U-3.02 on the 3rd floor of Building WSC of the Universität Duisburg Essen)

Vlada Limic (Strassburg): Large random graphs and excursions of Levy-type processes

• 30.10.2018 Birgit Jacob (Wuppertal): On continuity of solutions for parabolic control systems and input-to-state stability
Abstract: We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.

• 23.10.2018 M. Taeufer (Queen Mary, London): Die integrierierte Zustandsdichte fuer Laplaceoperatoren auf Archimedischen Gittern
Vortrag im Rahmen der 'Research Training Group High-dimensional Phenomena in Probability - Fluctuations and Discontinuity'.

• 16.10.2018 B. Richert (TU Dortmund): Sharpness of the phase transition in percolation

• 9.10.2018 (um 10:00 Uhr im Raum M614/616)
M. Taeufer (Dortmund/London): Quantitative Unique Continuation and Applications
Abstract: Verteidigung der Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften an der Fakultaet fuer Mathematik der Technischen Universitaet Dortmund.

• 08.10.2018 (Monday, time: 15:30, location: room WSC-S-U-3.02 on the 3rd floor of Building WSC of the Universität Duisburg Essen)

Abel Klein (University of California, Irvine): Manifestations of dynamical localization in the random XXZ quantum spin chain
Abstract: We study random XXZ quantum spin chains in the Ising phase. We prove droplet localization, a single cluster localization property that holds in an energy interval near the bottom of the spectrum. We establish dynamical manifestations of localization in the energy window of droplet localization, including non-spreading of information, zero-velocity Lieb-Robinson bounds, and general dynamical clustering. A byproduct of our analysis is that this droplet localization can happen only inside the droplet spectrum. (Joint work with Alex Elgart and Gunter Stolz.) Hosted by the Graduiertenkolleg 2131.

• 26.07.2018 (Thursday, time: 13:15, location: M 611) M.W. Leidgen(TU Dortmund): Irrfahren auf Perkolationsclustern

• 26.07.2018 (Thursday, time: 15:15, location: M 611) Ivica Nakic (University of Zagreb): Optimal control of parabolic equations using spectral calculus
Abstract

• 10.07.2018 Delio Mugnolo (FU Hagen): Comparison principles for parabolic equations and applications to PDEs on networks
Abstract: If a semi-bounded and symmetric but not essentially self-adjoint operator drives a PDE and boundary conditions have to be imposed and the corresponding solutions can often be compared: an efficient variational principle due to Ouhabaz shows e.g. that the solution to the heat equation with Neumann conditions dominates pointwise and for all times the solution of the heat equation with Dirichlet conditions and same initial data; whereas no such domination can hold if diffusion equations driven by two different elliptic operators under, say, Neumann b.c. are considered. In this talk I am going to discuss how domination theory can be extended to study domination patterns that only hold on long time scales: I will present (purely spectral!) criteria that imply either "eventual domination" or "interwoven behavior" of orbits of semigroups. Our main application will be given by heat equations on networks: remarkably, recently obtained results on spectral geometry for quantum graphs turn out to deliver prime examples where such criteria are satisfied.

• 03.07.2018 Martin Tautenhahn (TU Chemnitz): Tiling theorems and application to random Schrödinger operators with Gaussian random potential
Abstract

• 19.06.2018 Sebastian Fuchs (TU Dortmund, Statistik): Minimizer of Kendall's tau and Kendall's tau of the Order Statistic

• 12.06.2018 (15:00) Magda Khalile (Paris-Sud): Spectral asymptotics of Robin Laplacians on polygonal domains
Abstract

• 12.06.2018 (14:15) Thomas Ourmières-Bonafos (Paris-Sud): Dirac operators and delta interactions
Abstract: In this talk, we will discuss different aspects of the Dirac operator in dimension three, coupled with a singular potential supported on a surface. After motivating the study of such objects, we will briefly be interested in the problem of self-adjointness for singular electrostatic or Lorentz-scalar potentials. For this last class of potentials, we will study the structure of the spectrum of such an operator and in particular, we will show that for an "attractive" potential, when the mass of the particle goes to infinity, the behavior of the eigenvalues is given by an effective operator on the surface. We will see that this effective operator is actually a Schrödinger operator with both a Yang-Mills potential and an electric potential, each one being of geometric nature.
These are joint works with Markus Holzmann, Konstantin Pankrashkin and Luis Vega.

• 05.06.2018 (14:15) Alexander Dicke (Universität Siegen): Das Kakeya-Problem und Verbindungen zur Harmonischen Analyse

• 22.05.2018 (14:15) Johannes F. Brasche (TU Clausthal): Convergence of Schrödinger operators
Abstract, arXiv

• 08.05.2018 (14:15) Wolfgang Spitzer (FU Hagen): Entanglement entropy of the free Fermi gas at any temperature
Abstract: We consider the large-scale behaviour of the local entropy and the spatially bipartite entanglement entropy of thermal equilibrium states of non-interacting fermions in position space $$\mathbb R^d$$ ($$d\ge 1$$) at temperature, $$T\ge0$$. This leads to the study of the asymptotics of traces of non-smooth functions of Wiener-Hopf operators with smooth (at $$T>0$$) symbols and discontinuous symbols (at $$T=0$$). This is joint work work with Hajo Leschke and Alexander V. Sobolev.

• 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, room M1011) Max Kämper (TU Dortmund, Physik): 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): $$L_\infty$$-Abschätzungen 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

• 08.11.2016 & 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.

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Fakultät für Mathematik
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44227 Dortmund

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#### Sekretariat

Janine Textor (Raum M 620)

Tel.: (0231) 755-3063
Fax: (0231) 755-5219
Mail: janine.textor@tu-dortmund.de