Oberseminar Analysis, Mathematische Physik & Dynamische Systeme

A challenging mathematical problem is to introduce meaningful curvature notions on discrete spaces like graphs (which do not have any smooth structure). An analytic Ricci type curvature, based on Bochner's identity involving a Laplacian (or more generally the generator of a Markov semigroup) and going back to Bakry and Emery in 1985, was initially studied in the context of graphs by Elworthy, Schmuckenschlaeger, Lin/Yau and others. The first aim of this talk is to motivate and to introduce this curvature notion. We will then present a graph theoretical version of Buser's inequality. Buser's inequality links non-negative Ricci curvature, a fundamental isoperimetric constant named after Jeff Cheeger, and the first positive eigenvalue of the Laplace operator. Buser provided back in 1982 two proofs of his inequality (both of them geometric in flavour - one was using Geometric Measure Theory and the Heintze-Karcher inequality while the other was more elementary). An alternative analytical proof in the graph setting and based on the heat semigroup was later given by Klartag/Kosma/Ralli/Tetali in 2016. They utilized a heat semigroup reformulation of Bakry-Emery curvature and their proof is analogous to a corresponding analytical proof by Ledoux in 2004 in the manifold case. This is one of the many instances when the seemingly innocent heat equation is of central importance in completely different and new contexts. If everything goes according to plan, I will finish the talk with an optimal ratio estimate between higher eigenvalues and the first Laplace eigenvalue by combining an improved Cheeger inequality for graphs due to Kwok/Lau/Lee/Oveis Gharan/Trevisan (2013) with the above mentioned Buser inequality. This talk is a review of joint work with Shiping Liu (USTC, Hefei, China).

In the talk we discuss an arbitrary metric graph, to which we glue a graph with edges of lengths proportional to a small parameter $\varepsilon$. On such graph, we consider a general self-adjoint second order differential operator with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be holomorphic in $\varepsilon$. We introduce a special operator on a special graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that that certain parts of the resolvent of the original operator are holomorphic in $\varepsilon$ and we show how to find effectively all coefficients in their Taylor series. This allows us to represent the resolvent of by an uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of $\varepsilon$. In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in $\varepsilon$-expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident.

A function $k(x,y)$ of two variables $x>0$ and $y>0$ is called a Hardy kernel, if it is homogeneous of degree minus one: $k(ax,ay)=k(x,y)/a$. Hardy kernels are usually associated with integral operators on the positive semi-axis. Because of the homogeneity of the kernel, such operators are easily diagonalisable by the Mellin transform, and so the spectral theory of this class of operators is very simple. In the talk, I will discuss spectral properties of matrices $k(n,m)$, obtained by restrictions of Hardy kernels onto integers. Of course, after restricting to integers, the homogeneity property "disappears" and it is no longer clear how to diagonalise such matrices. Nevertheless, it turns out that some progress can be made in their spectral analysis. I will state some general theorems and consider in detail some examples.

We consider the ideal Fermi gas of indistinguishable, spin-less fermions confined to a Euclidean plane $\mathbb{R}^2$ perpendicular to an external magnetic field. We assume this (infinite) quantum gas to be in a ground state. For this (pure) state we define its entanglement entropy $S(\Lambda)$ associated with a bounded (sub)region $\Lambda\subset \mathbb{R}^2$ as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $\Lambda$ of finite area $|\Lambda|$. Assuming that the boundary curve $\partial\Lambda$ of $\Lambda$ is sufficiently smooth, we prove the leading asymptotic growth of $S(L\Lambda)$, as the scaling parameter $L>0$ tends to infinity. In the first talk we assume that the magnetic field is constant while in the second talk we extennd this to a magnetic field that is only asymptotically constant to $B_0$, say. Here, we show stability of the leading asymptotic growth of $S(L\Lambda)$ in the sense that it only depends on $B_0$. These results are in agreement with a so-called area-law scaling. It contrasts the zero-field case $B=0$, where an additional logarithmic factor $\ln(L)$ is known to be present. We mention the connection to Szegö asymptotics of Toeplitz matrices and we do not assume familiarity with (quantum) statistical mechanics.

We consider the ideal Fermi gas of indistinguishable, spin-less fermions confined to a Euclidean plane $\mathbb{R}^2$ perpendicular to an external magnetic field. We assume this (infinite) quantum gas to be in a ground state. For this (pure) state we define its entanglement entropy $S(\Lambda)$ associated with a bounded (sub)region $\Lambda\subset \mathbb{R}^2$ as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $\Lambda$ of finite area $|\Lambda|$. Assuming that the boundary curve $\partial\Lambda$ of $\Lambda$ is sufficiently smooth, we prove the leading asymptotic growth of $S(L\Lambda)$, as the scaling parameter $L>0$ tends to infinity. In the first talk we assume that the magnetic field is constant while in the second talk we extennd this to a magnetic field that is only asymptotically constant to $B_0$, say. Here, we show stability of the leading asymptotic growth of $S(L\Lambda)$ in the sense that it only depends on $B_0$. These results are in agreement with a so-called area-law scaling. It contrasts the zero-field case $B=0$, where an additional logarithmic factor $\ln(L)$ is known to be present. We mention the connection to Szegö asymptotics of Toeplitz matrices and we do not assume familiarity with (quantum) statistical mechanics.

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.

In this talk we discuss recent results on the asymptotic behaviour of the spectral gap of one-dimensional Schrödinger 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. Täufer).

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 $B\in L(U,X)$, wobei $U$ ebenfalls ein Banachraum ist. Wir studieren zum einen das Konzept Stabilisierbarkeit, d.h. Existenz eines $K\in L(X,U)$ 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, $C\in L(X,Y)$, 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.

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.

In this talk, we discuss some limitations of the quantum graph approach in defining Dirichlet forms, Laplacians and/or Brownian motions in case when the graph has vanishing edge lengths. Our model set is the stretched Sierpinski gasket, SSG for short. It is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. (It has also been called "Hanoi attractor".) As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing together the Brownian motions on the intervals. In fact, there have been several works in this direction. However, there still remains, "reminiscence" of the Sierpinski gasket in the geometric as well as analytic structure of SSG and the same should therefore be expected for diffusions. This is a joint work with Patricia Alonso Ruiz (Texas A&M University) and Jun Kigami (Kyoto University).

A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millennium, the mathematical understanding of this fact has progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent and partial progress in the direction of proving conformal invariance for a large class of such models.

Disordered systems such as glasses and spin glasses pose a challenge to theoretical physics. As a simplification mean-field models where the geometry of interactions is induced by a random graph have been introduced. These models turn out to be intimately related to deep questions in combinatorics that have been studied in their own right for well over half a century. Some of these questions have exciting applications in statistics and computer science. In this talk I am going to give an overview of this interdisciplinary research area and of the new contributions that physics insights have enabled.

Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in specific situations, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray-Scott equations constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear. In the derivation of a macroscopic model such as the deterministic Gray-Scott equations from basic physical principles, certain aspects of microscopic dynamics, e.g.~fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations. The randomness leads to a variate of new phenomena and may have a highly non-trivial impact on the behaviour of the solution. E.g. it has been shown by numerical modelling that the stochastic extension leads to a broader range of parameter with Turing patterns by a genetically engineered synthetic bacterial population in which the signalling molecules form a stochastic activator-inhibitor system. The stochastic extension may lead to multistability and noise-induced transitions between different states. In the talk, we will introduce the Gray Scott system, which is a special case of an activator-inhibitor system. Then, we introduce its numerical modelling and highlight the proof of convergence. References: Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray-Scott equations. J. Comput. Appl. Math. 364, Joint work with Jonas Toelle: A Schauder Tychonoff type Theorem and the stochastic Klausmeier system (Archive)

The Kardar-Parisi-Zhang (KPZ) universality class of stochastic growth models has several connections with other a-priori unrelated models in mathematics physics. We will describe which models belongs to the KPZ class, present some of the known result, and highlight some connections, in particular the one with random polymers and random matrices.

We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy-Widom distribution at the spectral edges of the Wigner ensemble. This is a joint work with Giorgio Cipolloni and Dominik Schroder.

There is no rigorous mathematical definition of a quasicrystal. In spaces with some group translation the latter term usually refers to well-scattered point sets (Delone sets) that are not periodic but display long range symmetries. Classes of these sets such as model sets have already been studied by Meyer in the 70's, i.e. some time before Shechtman's discovery of physical alloys with non-periodic molecular structure in 1982 (Nobel prize for chemistry in 2011). In this talk we focus on non-periodic point sets in lcsc groups (with the Heisenberg group as a guiding example) that are not too far from crystals in a dynamical sense. The main focus will be on the generalization of the concept of linearly repetitive Delone sets known from Euclidean space. Although they are not found easily, non-periodic, linearly repetitive Delone sets exist in many non-Abelian groups as well. Going beyond a result from Lagarias and Pleasants for R^d roughly outline how to prove unique ergodicity for the associated dynamical systems for a class of Lie groups of polynomial volume growth. Joint work with Siegfried Beckus and Tobias Hartnick.

Compressive Sensing is a recent development in mathematical signal processing which predicts that vectors that are approximately sparse can be accurately reconstructed from a number of linear measurements that is much smaller than previously believed possible. Efficient algorithms such as convex optimization approaches can be used for the reconstruction. Somewhat surprisingly random matrices model provably optimal measurement schemes. While the most accurate analysis of this phenomenon is available for Gaussian random matrices, practical applications demand for more structure in the measurement matrix. Structured random matrices of particular interest arise from randomly sampling the Fourier transform of the signal as well as subsampling the convolution of the signal with a random vector. We will discuss tools for analyzing the latter type of structured random matrices. It turns out that this leads to bounding the supremum of certain second order chaos processes. In particular, we will present a recent result that provides generic chaining type bounds via gamma-2-functionals. Joint work with Felix Krahmer and Shahar Mendelson.

In this note we investigate propagation of smallness properties for solutions to heat equations. We consider spectral projector estimates for the Laplace operator with Dirichlet or Neumann boundary conditions on a Riemanian manifold with or without boundary. We show that using the new approach for the propagation of smallness from Logunov-Malinnikova [7, 6, 8] allows to extend the spectral projector type estimates from Jerison-Lebeau [3] from localisation on open set to localisation on arbitrary sets of non zero Lebesgue measure; we can actually go beyond and consider sets of non vanishing $d - \delta$ ($\delta > 0$ small enough) Hausdorff measure. We show that these new spectral projector estimates allow to extend the Logunov-Malinnikova's propagation of smallness results to solutions to heat equations. Finally we apply these results to the null controlability of heat equations with controls localised onsets of positive Lebesgue measure. A main novelty here with respect to previous results is that we can drop the constant coefficient assumptions (see [1, 2]) of the Laplace operator (or analyticity assumption, see [4]) and deal with Lipschitz coefficients. Another important novelty is that we get the first (nonone dimensional) exact controlability results with controls supported on zero measure sets. (Joint work with N. Burq)

We consider the homogenization of Poisson and Stokes equations in a bounded domain of $\mathbb{R}^d$, $d > 2$, perforated by many small random holes. We assume that the holes are generated by properly rescaled balls having random radii and centers (i.e. Boolean process). Our main assumption is that the random radii of the Boolean process have finite $(d  2)$-moment: This condition is minimal in order to ensure that the average density of capacity generated by the holes is finite, but still allows for the onset of clustering balls with overwhelming probability. By combining analytic and probabilistic percolation-like methods, we give a homogenization result for a large class of measures as above. We prove that the homogenized equations are analogous to the case of periodic spherical holes. More precisely, for the Poisson equation, we recover in the homogenization limit an averaged version of the strange term established by Cioranescu and Murat; in the case of Stokes equations, we show that the homogenized solution solves a Brinkmann-type system. These are joint works with R.M. Höfer and J.J.L. Velázquez (University of Bonn).

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $X$.

In this talk we study sufficient conditions for observability 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 Dennis Gallaun (TU Hamburg) and Martin Tautenhahn (TU Chemnitz).

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.

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.

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 Günter Stolz.)

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.

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.

As it is well known convergence of lower semibounded closed quadratic forms implies convergence of the associated self-adjoint operators in the norm re- solvent sense. We derive a sharp estimate for the rate of convergence. If the sequence $(\mu_n)$ of finite signed Radon measures on $\mathbb{R}$ converges weakly to $\mu$, then the Schrödinger operators $\Delta+ \mu_n$ converge to $\Delta+ \mu$ in the norm resolvent sense. We sketch a proof and provide an upper bound for the rate of convergence with the aid of the mentioned general result. We give an algorithm for the numerical computation of negative eigenvalues of one- dimensional Schrödinger operators $\Delta+ \mu$, $\mu$ a finite signed Radon measure, and use above convergence results in order to derive error bounds. Finally we sketch the three-dimensional analogue. The results have been obtained joined with Robert Fulsche and Katarina Ozanova, respectively.

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 $\mathbb{R}^d$ ($d\geq 1$) at temperature, $T\geq 0$. 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.

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.

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).

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.

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.

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.

The Anderson model was invented 1985 by Anderson and describes the quantum mechanical motion of a particle in a random potential in $\mathbb{Z}^d$. 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).

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 $S_{xy}=E|H_{xy}|^2$. 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 that substantially improves the earlier bound $2\|S\|_\infty^{1/2}$ 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.

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.

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.

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.

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 $V\colon \Omega\times\mathbb{Z}^d\to\mathbb{R}$ is a Gaussian process with sign-indefinite covariance function.

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.

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 $\mathrm{div} h \mathrm{grad}$ 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).

Wir betrachten ein System, welches sich durch die Wärmeleitungsgleichung mit Neumann-Randbedingung auf einem glatten Gebiet im $\mathbb{R}^d$ 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.

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).

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 $n(r,w)=\mathcal{O}(r^2)$ 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.

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.

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.