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. Grubisic, V. Kostrykin, K. A. Makarov und K. Veselic

We consider simple parameter identification problems arising in continuum mechanics, focussing on a finite number of parameters and observations. Some error estimates and estimators for finite element discretisations are reviewed. We discuss implementation based on standard numerical optimization libraries.

In recent years, spacecraft navigation and control has achieved astonishing feats. As a recent example, the journey of ESA's Rosetta craft through the inner Solar System lasted ten years and covered 6.4 billion km until the lander Philae landed on comet 67P in November 2014. Likewise, the Japanese craft Hayabusa2 will fly 5.4 billion kilometers within six years in order to return samples from the asteroid Ryugu to Earth.
In this talk, we highlight some of the difficulties of planning corresponding flight paths of future missions in an optimal fashion, present some of the underlying mathematical models, and discuss methods from mathematical optimization that help to attain such amazing accomplishments.

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).

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.

Bei diesem Vortrag soll den Hörerinnen und Hörern leicht verständlich die Simulation von Extrusionsprozessen nahegebracht werden. Besonderer Wert wird auf die Simulation von Ein- und Doppelschneckenextrudern und die sich daraus ergebenden Konsequenzen für die Prozessoptimierung gelegt.
In einem zweiten Teil wird auf die Simulation von Werkzeugen und von Wendelscher- und Mischteilen eingegangen.
Abschließend wird die optimale Geometriegestaltung durch die Nutzung von KI-Systemen vorgestellt und die Anwendung im Bereich Extrusion erläutert.

In this talk we present a robust methodology for solving incompressible, immiscible two-phase flows modeled by the Navier-Stokes equations. All the equations are solved using continuous Galerkin finite elements. We start by describing the general algorithm, which employs a splitting operator technique that considers a velocity field to advect the material interface via a phase conservative level-set method. Afterwards, the new interface location is used to reconstruct the material parameters and to solve the Navier-Stokes equations to obtain a new velocity field. Most of the presentation will be devoted to the representation and time evolution of the interface. There is an extensive list of methodologies to treat material interfaces. Popular choices include the volume of fluid and level-set methods. We propose a novel level-set like model for multiphase flow that preserves the initial mass of each phase. The model combines and reconciles ideas from the volume of fluid and level-set methods by solving a non-linear conservation law for a regularized Heaviside function of the level-set function. By doing this, we guarantee conservation of the volume enclosed by the interface. Our level-set model contains a term that penalizes deviations from the distance function. The result is a non-linear monolithic model for a phase conservative level-set with embedded redistancing. To solve the Navier-Stokes equations we use a second order projection scheme. Using ideas for solving hyperbolic PDEs via continuous Galerkin finite elements, we propose a robust and parameter free stabilization for the momentum equations. This stabilization is suitable for unstructured meshes and has been tested for multiple refinement levels. We present several numerical examples to demonstrate the behavior of this method under different scenarios.

Recent advances have enabled the development of efficient high-order entropy-stable discretizations for the Euler and Navier-Stokes equations. However, the strict entropy-stability property is destroyed by standard explicit time discretizations. I will present a class of Runge–Kutta-like methods, related to projection methods, that guarantee conservation or stability with respect to any inner-product norm, and thus provide fully-discrete entropy stability for symmetric hyperbolic systems at the same cost as standard explicit Runge-Kutta time stepping. Because of the methods’ special form, they retain many desirable properties (including order of accuracy, approximate linear stability, and strong stability preservation) of the original Runge–Kutta method. I will show several numerical examples, including an extension to preservation of stability for arbitrary convex entropies such as the standard entropy for the Euler equations.
This is joint work with H. Ranocha, M. Alsayyari, M. Parsani, and L. Dalcin.

Multi scale analysis is a method to prove spectral and dynamical localisation of wavepackes whose motion is governed by a random operator of Schroedinger type. To be able to apply this method one has to ensure that certain assumptions are satisfied, which physically correspond to an appropriate energy x disorder regime. Oftentimes the random Schroedinger operator contains an explicit tuning parameter which couples the random perturbation. The talk discusses appropriate starting assumptions for the multi scale analysis for models including an explicit disorder parameter, which however enters the model possibly in a non-linear way.

The inverse problem of resolving the 'cocktail party problem' is the topic of discussion in this lecture. As an example, consider listening to a beautiful piece of classical music either from playing a CD or enjoying a symphony orchestra. The sound of the music being played is considered a blind source. The blind-source recovery problem is the inverse problem of identifying which musical instruments are being played, the number of all such instruments, as well as finding the amplitudes, musical tone, and time-varying (or instantaneous) frequencies, produced by each individual musical instrument. Being able to identify the music produced by every individual instrument requires determining the times of 'arrival (signal onset) and departure (signal offset)' of the sound produced by this particular instrument and calculating its sound volumes (amplitudes) and instantaneous frequencies (IFs).
We will first introduce a general mathematical model of composite signals and time series, and explain why the existing decomposition methods fail to resolving the inverse problem. We will then recall two existing and somewhat successful approaches, namely: 'synchro-squeezing transform (SST)' proposed by Ingrid Daubechies, and 'signal separation operation (SSO)' introduced by Hrushikesh Mhaskar and myself, while pointing out that they are completely different but sharing the same requirement of positive IFs in the entire time interval and the necessity of adjusting two (free) parameters for extracting the IFs, from which the composite signal or time series components are to be resolved.
A new theory, along with effective computational schemes and algorithms, based on spline approximation and continuous wavelet transform (CWT), will be presented in this lecture. As a departure from the SST and SSO approaches, extracting the IFs is not necessary for resolving the inverse problem of separating and recovering the components of the composite signal or time series from the blind source. Instead, the scale of the CWT is the only parameter to be determined, for which wavelet thresholding is instrumental to separating this wavelet scale into clusters, with cardinality to match the number of (active) components that constitute the composite signal or time series, and that extrema estimation for each cluster gives rise to the optimal scales, from which the components as well as their corresponding IFs are recovered independently.

Bernstein's theorem that an entire minimal graph over the Euclidean plane is flat is a key result in the theory of nonlinear partial differential equations, and the challenge to generalize this to higher dimensions and codimensions became a guiding problem for geometric analysis in the second half of the 20th century. The lecture will present recent discoveries that lead to a more complete picture.

Copulas are the link between multivariate distributions and their univariate marginals, studying dependence therefore means studying copulas. Complementing their obvious importance for statistics and data analysis in general, copulas - or, equivalently, d-stochastic measures - are also interesting objects from the perspective of geometry (characterizing extreme points), from the perspective of topology (peculiar sets are co-meager), and from the perspective of dynamical systems (iterates of the so-called star product, iterations of shuffles, etc.). Additionally, despite intuitively being regular objects copulas can exhibit surprisingly singular behavior and can, e.g., have fractal supports. The talk first introduces copulas and d-stochastic measures and then discusses both solved as well as open problems illustrating the afore-mentioned aspects.

Abstract: Standard dependence measures considered in the (mostly non-mathematical) literature like Pearson correlation, Spearman correlation, the Maximal information coefficient (MIC), and Schweitzer and Wolff’s famous sigma are symmetric, i.e. they assign each pair (X, Y) of random variables the same dependence as they assign the pair (Y, X). Independence of two random variables is a symmetric concept modelling the situation that knowing X does not yield any information gain about Y and vice versa - dependence, however, is not. Thinking, for instance, of a sample (x1, y1), ..., (xn, yn) roughly in the shape of a noisy letter V, it is without doubt (on average) easier to predict the y-value given the x-value than vice versa.
In 2010 a Markov kernel based, scale-free dependence measure quantifying directed dependence of pairs of random variables was introduced in [1]. In 2018 a checkerboard-based estimator for this dependence measure was derived and implemented in the R-package qad (short for quantification of asymmetry in dependence). Given a sample sample (x1, y1), ..., (xn, yn) the R-package estimates the dependence of the second variable on the first one and vice versa, and, as a byproduct, calculates the asymmetry in dependence of the underlying dependence structure.
After recalling some background on copulas the talk will sketch the basic ideas behind the qad-estimator, show that the estimator is strongly consistent (without any regularity assumptions), and illustrate its asymptotic as well as its small sample properties by some examples.

[1] W. Trutschnig: On a strong metric on the space of copulas and its induced dependence measure, Journal of Mathematical Analysis and Applications 384, 690-705 (2011), doi:10.1016/j.jmaa.2011.06.013

Many mathematical CFD models involve transport of conserved quantities that must lie in a certain range to be physically meaningful. The solution u of a scalar conservation law is said to satisfy a maximum principle (MP) if global bounds u_min and u_max exist such that u_min <= u(t,x) <= u_max. To enforce such inequality constraints for DG solutions at least for element averages, the numerical fluxes must be defined and constrained in an appropriate manner. We introduce a general framework for calculating fluxes that produce non-oscillatory DG approximations and preserve the MP for element averages even if the exact solution of the PDE violates them due to modeling errors or perturbed data. The proposed methodology is based on a combination of flux and slope limiting: The flux limiter constrains changes of element averages so as to prevent violations of global bounds. The subsequent slope limiting step adjusts the higher order solution parts to impose local bounds on pointwise values of the high-order DG solution. Since manipulations of the target flux may introduce a consistency error, it is essential to guarantee that physically admissible fluxes remain unchanged. The novel fractional step flux limiting approach is iterative while in each iteration, the MP property is guaranteed and the consistency error is reduced. Practical applicability is demonstrated by numerical studies for the advection equation (hyperbolic, linear) and the Cahn-Hilliard equation (parabolic, nonlinear) for which additionally some extensions are discussed. The flux limiter (similar to a slope limiter) is a local/parallelizable postprocessing procedure that can be applied to various types of DG discretizations of a wide range of scalar conservation laws.

Der Übergang von der Schule an die Hochschule stellt Lernende häufig vor große Herausforderungen. Die Hochschulen haben hierauf in den letzten Jahren mit vielfältigen Maßnahmen (z. B. durch Eingangstests, Vor- und Brückenkurse sowie semesterbegleitende Angebote) reagiert. Neben dem Vorwissen, das Studierende an die Hochschule mitbringen, wird die Leistung im Studium von weiteren Faktoren beeinflusst. So haben etwa Selbstwirksamkeitserwartungen Einfluss auf die Studienleistung. Zur Gestaltung des Übergangs werden Maßnahmen für einen nachhaltigen Mathematikunterricht, für die Konkretisierung der Bildungsstandards, für die Gestaltung des Übergangs Schule–Hochschule und die Mathematikausbildung im Studium gefordert. Beispiele für konkrete Unterstützungsangebote zeigen Chancen und Möglichkeiten im Umgang mit dem Übergang Schule-Hochschule, und empirische Ergebnisse verdeutlichen mögliche Zusammenhänge zwischen Vorwissen, Maßnahmen zu Studienbeginn und Studienerfolg.

Unique continuation principles constitute a very active field in control theory or the theory of random Schroedinger 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.