Oberseminar Analysis, Mathematische Physik & Dynamische Systeme

The Lusternik-Schnirelmann category of a space gives a lower bound on the number of critical points of a continuously differentiable function on a Banach manifold. Related to Lusternik-Schnirelmann category is the concept of topological complexity. This is a numerical homotopy invariant and is of great interest in topological robotics. As extensions of topological complexity one considers the notions of sequential and parametrized topological complexity. In this talk we introduce topological complexity and show how one can apply methods from Lusternik-Schnirelmann theory to establish various lower bounds on numbers of critical points of functions in terms of sequential and parametrized topological complexities. We also discuss applications of these results. This talk is based on joint work with Stephan Mescher.

We provide norm-resolvent estimates in various operator norms, for the class of problems in linear elasticity describing thin heterogeneous rods with rapidly oscillating coefficients in the regime of moderate contrast. The estimates are provided with respect to the period of material oscillations in the setting of simultaneous homogenization and dimension reduction, while assuming that the period and the rod thickness are of the same order. These estimates are expected to provide sharp estimates for the corresponding evolution problems. The analysis is performed by the means of spectral analysis and Gelfand transform. This is joint work with K. Cherednichenko and I. Velčić

We investigate the connection between the propagation of smallness in two dimensions and one-dimensional spectral estimates. By revisiting Zhu’s proof of the propagation of smallness in the plane, we obtain a quantitative version that includes an explicit dependence on key parameters. This refinement enables us to establish spectral inequalities for one-dimensional Schrödinger operators.

Consider an open set $\Omega \subset \mathbb{R}^N$ and the operator $Q_\Omega^\alpha$ acting as the Laplacian $u \mapsto -\Delta u$ on $\Omega$ with the Robin boundary condition $\partial_\nu u = \alpha u$ on $\partial \Omega$, where $\partial_\nu$ is the outward normal derivative and $\alpha > 0$ is the so called Robin parameter. We are interested in the strong coupling asymptotics of the Robin eigenvalues $E_j (Q_\Omega^\alpha)$. First we discuss the cases where $\Omega$ has a smooth/Lipschitz boundary or has an isotropic peak. After that we move on to non-isotropic peaks. More precisely: Let $\Omega \subset \mathbb{R}^3$ be an open set such that \begin{align*} \Omega \cap \{x_3 < \delta\}=\left\{(x_1, x_2, x_3) \in \mathbb{R}^2 \times (0,\delta) \colon \left( \frac{x_1}{x_3^p},\frac{x_2}{x_3^q} \right) \in (-1,1)^2 \right\} \subset \mathbb{R}^3, \ \Omega \setminus \{x_3 \leq \delta\} \ \text{is a bounded Lipschitz domain,} \end{align*} for some $\delta > 0$ and $1 < p < q < 2$. If a set satisfies the first condition one says that it has a non-isotropic peak at 0. We will take a glimpse on how to prove that for large $\alpha$ the $j$th eigenvalue behaves as $E_j (Q_\Omega^\alpha)\approx \mathcal{A}_j \alpha^{\frac{2}{2-q}}$, where the constants $\mathcal{A}_j < 0$ are eigenvalues of a one dimensional Schrödinger operator which depends on $p$ and $q$.

Im Kontrast zur derzeitig dominanten Wahrnehmung der Mathematik als 'Schlüsseltechnologie' wird der Vortrag nach dem bildenden Wert der Mathematik jenseits ihrer direkten Anwendbarkeit fragen. Dabei werden u.a. drei Positionen skizziert: Platon als Ausgangspunkt der Überlegungen zur Mathematik im Bildungsgang und zugleich als Höhepunkt der antiken Sicht, Nikolaus Cusanus als Protagonist des christlichen Mittelalters und zugleich als Türöffner der Moderne und schließlich als klassische Repräsentanten der Modernen Mathematik, Karl Weierstraß, David Hilbert und Hermann Weyl.

The talk is aimed to present a new kind of symmetry; namely scaling symmetries in the context of symplectic-Hamiltonian systems. These symmetries allow a geometrical reduction of the system into a scale-invariant one. The mathematical framework takes place in the realm of symplectic, and contact geometry- the odd-dimensional cousin of symplectic geometry. The talk will briefly present the former geometry and define the Hamiltonian dynamics in such a context. The main result of the talk will show exactly how the dynamics of a scale-dependent system can be projected onto the scale-independent one. The generality of this formalism can be widely applied, however, its use becomes conceptually obvious when applied to Big Bang cosmological models, since they encounter singularities when the scale of the system approaches zero or infinity. This last application will be briefly discussed if time allows it.

This talk is about the spectral properties of the operator T defined by the non-selfadjoint discrete Klein Gordon equation with eigenparameter dependent boundary condition. Spectral singularities are defined as poles of the resolvent?s kernel which are embedded in the continuous spectrum, and they are not eigenvalues. Naimark was the first to investigate the quantitative properties of the eigenvalues and spectral singularities of the Sturm-Liouville differential operator. Due to its wide applications in physics, spectral analysis of the Sturm-Liouville problem whose boundary conditions depend on spectral parameters have been a popular research topic. Inspired by the recent papers, we determine the conditions under which the eigenvalues and spectral singularities of the operator T has finite number of eigenvalues with finite multiplicities. This is based on joint work with Nihal Yoku? (Selçuk University, Türkiye).

TBA

We consider a weighted discrete graph $b$ over $(X,m)$, i.e. $X$ is a countable discrete set, $m$ is a function on $X$ which induces a measure and $b$ is an edge weight. Then the corresponding Laplacian is a non-negative self-adjoint operator. We investigate whether (weak) observability estimates for the corresponding heat equation exist, i.e. we study if, for given final time $T>0$ and a subset $D$ of $X$, the norm of the solution of the heat equation at time $T$ can be bounded by the portions of the solution on $D$ up to time $T$ and the norm of the initial condition. We will also comment on the impact of such estimates w.r.t. controllability. This is joint work with Peter Stollmann (Chemnitz) and Martin Tautenhahn (Leipzig).

This talk is about new so-called Bernstein-type inequalities. They are closely related to the Landau operator or “twisted Laplacian” which describes a two-dimensional Schrödinger operator with a constant magnetic field. We will explain how to prove magnetic Bernstein inequalities and how they can be used to deduce quantitative unique continuation principles for the Landau operator. This is based on joint work with Paul Pfeiffer (FernUniversität in Hagen)

Es werden quantitative Spektralungleichungen für (anisotrope) Shubinoperatoren auf dem Ganzraum diskutiert, die eine explizite Abhängigkeit von Abfallraten der Sensormenge aufweisen. Dies erweitert kürzlich erzielte Ergebnisse von J. Martin und erlaubt die Anwendung auf das Kontrollproblem für gewisse Baouendi-Grushin-Operatoren. Der Vortrag basiert auf einer gemeinsamen Arbeit mit P. Alphonse.

The Hardy inequality stands in the core of mathematical physics as having a Hardy inequality is strongly connected to hyperbolic manifolds and graphs, transient jump processes and subcritical energy functionals. Thus, it connects geometry, probability theory and analysis in a certain sense. Besides studying the validity of the inequality, the discussion of the optimality of the corresponding weights is of main interest. In this talk, we introduce the corresponding quasi-linear setting on locally finite graphs and show a method to obtain optimal Hardy weights. We illustrate this method on various examples. Moreover, we show a decay condition for Hardy weights. If the time permits, we also discuss characterisations of having a Hardy inequality in the quasi-linear setting.

The talk deals with the eigenvalue decay of solution operators to operator Lyapunov equations, a relevant topic in the context of model reduction for parabolic control problems. We mainly focus on the Gramian operator that arises in the context of control and observation of heat processes in infinite time. By improving existing energy and observability estimates for parabolic equations, we obtain both upper and lower bounds on the convergence rate of the eigenvalues of the Gramian operator towards zero. Both bounds follow the same polynomial decay rate, up to a multiplicative constant, which ensures their optimality. This confirms the slow decay of the eigenvalues and limits the efficiency of model reduction. The theoretical findings are supported by numerical results.

In this talk, I will present the Bernstein inequality and a spectral inequality for the two dimensional Landau Hamiltonian in a constant magnetic field. The best part is that the dependency on the chemical potential $\mu$, the magnetic field B and the thick set S are explicit. We might come to further topics regarding the Landau Hamiltonian at the end, if there is time left.

Wir betrachten die stationäre Diffusionsgleichung $-\nabla \cdot[a(\omega)\nabla u] =f$ in $D$ mit einem zufälligen Koeffizient $a=a(\omega)$ im Zusammenhang eines Randwertproblems mit Dirichlet -und Neumannbedingung. Dabei wird der reellwertige Koeffizient $a$, wie auch schon in anderen Arbeiten, durch ein Levy-Zufallsfeld $Z$ generiert. Das Levy-Zufallsfeld $Z$ soll dazu von Natur aus positiv sein, um es direkt als Koeffizient in der Diffusionsgleichung verwenden zu könne. Dies führt uns zu Compound Poisson Zufallsfeldern. Insbesondere wird der Zusammenhang zwischen den Daten, die die Verteilung des Feldes $Z$ bestimmen und der eindeutigen Lösbarkeit des Randwertproblems diskutiert.

This talk will present some details about the application of a theorem from empirical process theory to the quantification of the convergence of averaged sums of Eigenvalue-counting functions of a random operator with finite eigenvalues to the expected value of the Eigenvalue-counting function of this operator.

What can one say about the total ``energy'' $\|x(T)\|_X^{}$ of a system's state on a Banach space $X$ at a final time $T > 0$ given only incomplete observations $\|y(t)\|_X^{}$ for $t \in E \subseteq [0, T]$? Our answer to this question consists of providing a so-called $\textit{final-state observability estimate}$ \begin{align*} \|x(T)\|_{X}^{} \lesssim \int_E \; \|y(\tau)\|_X^{} \; \mathrm{d} \tau, \end{align*} i.e., estimating the final state's energy by aggregation of observations. More precisely, we present an extension of the $\textit{Lebeau--Robbiano strategy}$ to the setting of non-autonomous Cauchy problems on general Banach spaces featuring time-dependent families of bounded observation operators. This strategy builds on the existence of a suitable $\textit{uncertainty principle}$ and $\textit{dissipation estimate}$. As application, we consider differential operators on $\mathrm{L}^p(\mathbb{R}^d)$ and observation operators $C(t) = \mathbf{1}_{\Omega(t)}$ representing families of $\textit{moving sensor sets}$. More concretely, we show that a final-state observability estimate holds for the case of an evolution induced by a non-autonomous family of $\textit{strongly elliptic differential operators}$ or non-autonomous $\textit{Ornstein--Uhlenbeck operators}$ subject to a $\textit{generalized Kalman rank condition}$. For these examples, we will see that the existence of a final-state observability estimate is closely related to geometrical requirements on the family of sensor sets $(\Omega(t))$. This talk is based on joint work with C. Bombach, C. Seifert, and M. Tautenhahn, arXiv:2203.08469, to appear in SIAM Journal on Control and Optimization.

A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure showing that with high probability the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, in a regime where the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained. Arxiv: http://de.arxiv.org/abs/2206.03444

Two omnipresent concepts in the context of regression are predictability and explainability. Predictability involves quantifying the extent of functional (or monotone regression) dependence of the endogenous variable $Y$ on a set of $d \geq 1$ potential exogenous variables $X_1,\ldots ,X_d$, mostly with the aim to select the most relevant explanatory variables. Explainability refers to the quantification of the fraction of explained variance and quantitatively assesses the appropriateness of a chosen statistical model. Choosing the "simple measure of conditional dependence" T recently introduced by Azadkia & Chatterjee (2021) as a measure for predictability and the multivariate extension of copula correlation as a measure for explainability allows to work with a dimension reduction principle. We identify and investigate the dependence structure underlying this dimension reduction principle, provide a strongly consistent estimator for the aforementioned quantities, and demonstrate its broad applicability.

Biran-Cieliebak conjectured in 2001, that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In this talk, I will present a proof.

The evolution in time of a quantum system is characterized by the Hamiltonian. The talk will focus on the Hamiltonian related to the N-body problem. After introducing some notations and context for the N-body problem, I will present a result on the decomposition of the essential spectrum of a class of operators that contained the Hamiltonian. This result is a generalization of the well-known HVZ-theorem. This decomposition is obtained using the theory of algebra of operators with, in particular, the use of C*-algebras. In a second step, I will give results on the regularity of the eigenfunctions of the Hamiltonian. This regularity result is obtained using the theory of differentials geometry, with a focus on the blow-up of manifolds with corners. I will end with a result connecting these two approaches: the operators algebra's one and the differential's geometry one's. This connecting result show that these two approach are actually built on the same space. These are joint work with B. Amman, N. Prudhon and V. Nistor.

This talk deals with a general method to prove final-state observability estimates based on the Lebeau-Robbiano strategy. Complementing previous results obtained in the context of Hilbert spaces, we derive observability estimates of operator semigroups in general Banach spaces by combining an uncertainty principle and a dissipation estimate. Thereby, we allow for generalized growth rates in the assumptions and use an integral expression to estimate the observability constant. We study connections to controllability and stabilization properties and apply our results to linear parabolic equations with thick control sets.

Zunächst erläutern wir ein Eindeutigkeitskriterium für Energieminimierer zu quadratisch gleichmäßig konvexen Funktionalen in einer Klasse von geeigneten maßerhaltenen Abbildungen und bezüglich geeigneter Randbedingungen. Danach diskutieren wir Verallgemeinerungen und mögliche Anwendungen. Besonderen Fokus legen wir dabei auf eine Situation, die einen singulären Energieminimierer erlaubt.

The Poisson hyperplane tessellation in Euclidean space is a classical and well-studied model of Stochastic Geometry. In this talk we will consider its counterpart(s) in hyperbolic space. As it turns out, Euclidean hyperplanes have more than one analogue in hyperbolic space, corresponding, roughly speaking, to thinking of hyperplanes as either totally geodesic hypersurfaces, equidistant sets from other hyperplanes, or spheres of infinite radius. In hyperbolic space these notions give rise to a one-parametric family of hyperplane-like hypersurfaces, known collectively as lambda-geodesic hypersurfaces, for lambda between 0 and 1. We will consider the isometry-invariant Poisson process of lambda-geodesic hyperplanes, and in particular its total surface area within a growing spherical window. We will study its limit theory, and prove central and non-central limit theorems, depending on the value of lambda and the ambient dimension. Based on joint work with Z. Kabluchko and C. Thäle

In this talk, we will focus on the evolution equations associated with nonselfadjoint quadratic differential operators. On the one hand, it will be explained how the non-commutation phenomena between the selfadjoint and skew-selfadjoint parts of these operators allow the evolution equations studied to enjoy smoothing and localization properties in some specific directions of the phase space, that will be precisely described. On the other hand, it will be found that the properties of null-controllability of the Ornstein-Uhlenbeck equations (which are quadratic evolution equations) are related to a notion of integral thickness condition associated to the skew-adjoint part of the operators involved. These different properties will be deduced from a fine description of the polar decomposition of the evolution operators associated to the studied equations. These results come from a series of works with J. Bernier (LMJL) and J. Martin (IRMAR).

Nonlinear infinite-dimensional systems are nonlinear dynamical systems whose state components lie in an infinite-dimensional space, typically a function space. Such systems, which are also called distributed parameter systems, are ubiquitous in real-life since they are able to model many physical processes, going from conservative mechanical systems to dissipative phenomena. A lot of questions may arise when dealing with such classes of systems. For instance, the well-posedness in terms of existence and uniqueness of solutions as well as the study of the equilibria, their stability and their control are paramount steps when studying these dynamical systems. For that class of systems, we study the field of tracking control, and especially funnel control, which constitutes an appropriate tool for the output of a system to track a class of reference signals. As a main contribution on this topic, we extend the available results that allow to consider linear infinite-dimensional systems as internal dynamics to the nonlinear setting. We prove that a class of nonlinear infinite-dimensional systems that satisfy some standard assumptions admits a differential relation between the input and the output that is conducive for funnel control. Our theoretical results are illustrated by means of examples and numerical simulations.

Copulas and doubly stochastic measures are mostly well known in the statistical community due to Sklar's famous theorem saying that copulas are the link between multivariate distribution functions and their marginals. Copulas, however, exhibit various interesting and surprising mathematical aspects going far beyond statistics. The talk will discuss some of those aspects, in particular some interrelations with functional analysis, geometry, number theory, and fractals.

In this talk, we present recent mathematical results in Maxwell variational inequalities with technological applications in high-temperature superconductivity and EM-shielding. The established well-posedness opens a way to analyze the corresponding numerical approximations, optimal design and inverse problems, which are mainly complicated by the hyperbolic Maxwell structure and the non-smooth character. Results in these directions are presented.

We consider a sequence of linear-quadratic optimization problems defined in an abstract functional framework. Each problem is accompanied by the constraint of reaching a given target with some precision. We show that problems are well posed and we characterise their solutions. The main result provides the conditions under which these solutions converge to the minimizer of the limit problem. The theory can be applied to a wide range of problems: elliptic, parabolic, control ones etc. In the talk, the following examples will be presented: 1. optimal controllability of the heat equation 2. optimal solvability of the Poisson equation.

Decay properties of eigenfunctions of Shubin-type operators, i.e., Schrödinger operators with potentials of the form $|x|^\tau$ with $\tau > 0$, are examined. After that, a spectral inequality for these operators with sensor sets of finite Lebesgue measure is shown and put into context. The presented results are based on joint work with A. Seelmann and I. Veselić.

This talk will introduce the concept of almost-additive functions on lattices with the special case of eigenvalue-counting functions of random Schrödinger operators and showcase how they can be used in conjunction with some results from empirical process theory to find explicit error estimates for their convergence to the integrated density of states. This talk is based on joint work with Christoph Schumacher, Fabian Schwarzenberger and Ivan Veselić

The main topic of the talk are dynamical processes where the dynamics is occurring on a set of entities (nodes in a network) that are connected to each other via edges in some nontrivial way. This leads to a natural question of how such underlying nontrivial connectivity affects dynamical processes. We are in particular interested in the following question: if one is to disturb one or more nodes in order to disturb the dynamics of the whole network, which node(s) to choose? As our performance measure we use the concept of $H_\infty$ norm from control theory and analyse a class of networks where the dynamics of nodes are modeled by a 2nd order differential operator. We apply abstract results to the case of a wave equation on a metric graph. The talk is based on joint works with A. Bátkai, M. Kramar-Fijavž, I. Palunko, D. Tolić and Z. Tomljanović.

Results on spectral inequalities based on techniques from complex analysis are reviewed. Special emphasis is paid to the pure Laplacian on Euclidean domains and (partial) harmonic oscillators on $\mathbb{R}^d$, recently discussed in joint works with M. Egidi, A. Dicke, and I. Veselić.

In this talk, I will present new uncertainty principles holding in general Gelfand-Shilov spaces. We will focus on standard Gelfand-Shilov spaces $S^\mu_\nu$ where the parameters $\mu, \nu > 0$ satisfy $\mu + \nu \geq 1$. From these uncertainty principles, we will deduce sufficient geometric conditions for the null-controllability of evolution equations whose adjoint system enjoys Gelfand-Shilov smoothing effects. More precisely, these results describe how the control support is related to the two parameters $\mu$ and $\nu$ associated to the Gelfand-Shilov space $S^\mu_\nu$. In particular, they applies for evolution equations associated to fractional anisotropic Shubin operators.

Resonances of wave equations can be defined as complex poles of analytic continuations of spatial-cut-off resolvents. Roughly speaking, they describe the localization on bounded inhomogenieties for large times and can be seen as a type of dissipation eigenvalues. Asymptotics of resonances is usually studied by various means like the counting function for growing discs, resonance-free regions, and asymptotics of counting functions in specially shaped complex strips. After a short discussion of differences in these descriptions and of open problems for well-established theories of obstacle scattering and Schrödinger equations, it is planned to report recent results on resonances produced by random and deterministic families of point interactions following joint papers of Sergio Albeverio and the speaker. If time allows us, it is also planned to discuss briefly other approaches to random resonances.

In this talk, we will consider inverse problems for parabolic systems with dynamic boundary conditions. Our main aim is to prove stability estimates for the determination of some unknown parameters appearing in the system, namely, source terms, potentials and initial data, from internal measurements. First, we prove a Lipschitz stability result for a class of source terms by establishing a suitable Carleman estimate. Then we study a nonlinear inverse problem involving the recovery of two radiative potentials simultaneously with initial temperatures. We prove a Lipschitz stability estimate for the potentials using Carleman estimates along with some positivity results of the solutions. Then we infer a logarithmic stability result for the initial temperatures by a logarithmic convexity method. Afterwards, we prove a general logarithmic stability result for initial data in an abstract parabolic system. To do so, we use a general logarithmic convexity result. As an application, we consider the Ornstein-Uhlenbeck equation. Then, we will study the identification of source terms from the final overdetermination in the heat equation with dynamic boundary conditions. For this purpose, we combine a weak solution approach and a gradient method with an adjoint problem. We perform some numerical simulations for the reconstruction of an unknown source term in the 1-D equation. Finally, some conclusions and possible extensions will be mentioned.

We discuss observability estimates for autonomous parabolic PDE with constant coefficients such as the heat equation. The study of these estimates is motivated by the question of null-controllability of the dual equation. We will show how to obtain these observability estimates on $L^p$-spaces for any $p \in [1,\infty]$. In particular, this implies approximate null-controllability of the dual equation in $L^1$. In the second half of the talk, we will discuss an extension to nonautonomous equations.

In the 80s, the physicists Rammal and Tolouse observed that suitable series of eigenvalues in the finite graph approximations of the Sierpinski gasket produced an orbit of a particular dynamical system. That observation lead to a complete description of the spectrum of the standard Laplace operator by Fukushima and Shima. The study of this spectrum has since then revealed structures with many interesting features not seen in other more classical settings. For instance, it presents large exponential gaps (or spacings), whose existence and properties have extensively been studied. What happens with small gaps? This fairly challenging question had eluded previous investigations and is the main subject of the present talk, where we discuss yet another remarkable fact: Any two consequent eigenvalues in the Dirichlet or in the Neumann spectrum of the Laplacian on the Sierpinski gasket are separated at least by the spectral gap.

Information geometry suggests the Fisher-Rao metric for learning in deep neural networks, leading to the so-called natural gradient method. This method turns out to be extremely efficient in many applications, such as supervised or reinforcement learning. However, there is an ambiguity in the context of learning systems that consist of visible and hidden units. We have to consider two different natural geometries. One geometry is related to the full system, the other one to the visible sub-system. In principle, these two geometries imply different natural gradients. We compare them and prove an invariance property that distinguishes the Fisher-Rao metric from other Riemannian metrics based on Chentsov's classical characterisation theorem.

The philosophy of Agmon says that generalized eigenfunctions "which do not grow too fast in fact decay rapidly". Our setting are Schrödinger operators on graphs and the spectral regime is below the essential spectrum of the operator. The analysis is based on recent work on Hardy and Rellich inequalities on graphs. (This is joint work with Felix Pogorzelski.)

The Covid-19 pandemic still causes severe impacts to the society and the economy. This paper studies excess mortality during the pandemic years 2020 and 2021 in Germany empirically, with a special focus for the life insurer's perspective. Our conclusions are based on official counts of German governmental offices on the living and deaths of the entire population.

In group testing, a population is tested for a certain feature (i.e. an infection) by testing individual samples in pools. We give a brief overview to group testing and show how concepts from combinatorics such as Steiner Systems and in particular constructions based on Reed-Solomon-Codes lead to practical group testing strategies which can improve throughput, accuracy, and detection time. Based on joint work with Christoph Schumacher.

The impact of the Pauli principle, as expressed by the antisymmetry of many-particle wave functions, depends on the regularity of Sobolev function and hence on the space dimension, right? - Yes and no! On the one hand, all forms of so-called contact interactions are excluded in all space dimensions d >1 but not for d=1. On the other hand, the extent to which short-range forces of a certain normalized strength are weakened by the Pauli principle is independent of the space dimension. This is the result of a surprising interplay between spectral theory and Sobolev regularity. - Joint work with Michael Hofacker.

Introducing an interpolation method we estimate the spectral gap for Brownian motion on general domains with sticky-reflecting boundary diffusion associated to the first nontrivial eigenvalue for the Laplace operator with corresponding Wentzell-type boundary condition. In the manifold case our proofs involve novel applications of the celebrated Reilly formula. (Joint work with Victor Marx and Vitalii Konarovskyi.)

The Euler and the Navier-Stokes equations describe the motion of fluids and a smooth solution can be ensured only for small times or large viscosites. It is open if a smooth solution existence for all times or if a break down can appear in finite time. In the formulation of the problem the initial data shall be Schwartz functions. We show in arXiv:1912.11705 that then the vorticity remains a Schwartz function as long as the smooth solution exists. We provide explicit bounds for all seminorms. This provides new break down criteria for the Euler and the Navier-Stokes. Our approach is a Montel space based splitting method since the set of Schwartz functions is a Montel space. Contrary to other (especially Sobolev and weak solution) methods our approach is not affected by the viscosity, i.e., the result holds for the hyperbolic Euler and the parabolic Navier-Stokes equations.

A challenging mathematical problem is to introduce meaningful curvature notions on discrete spaces like graphs. 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. 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.