This talk is devoted to the large-scale rheology of systems of suspended particles in a Stokes fluid. After reviewing recent results on the definition of the effective viscosity of such systems in terms of homogenization theory, I will focus on its asymptotic expansion in dilute regime. I will present a new, optimal proof of Einstein's viscosity formula for the first-order expansion and show how to pursue the asymptotic expansion to higher orders. The essential difficulty originates in the long-range nature of hydrodynamic interactions: these require suitable renormalizations, which will be systematically captured by means diagrammatic expansions. This is based on joint work with A.Gloria.

For a (not necessarily convex) polygon with n vertices in the projective plane we consider the intersection points of the lines containing the edges, which are not the vertices. If no three of these lines have a common point, then there is a unique curve of degree n-3 going through these residual points, called the adjoint curve. If the polygon is real and convex, then the adjoint curve is a hyperbolic curve. For a given hyperbolic cubic we give a method constructing all associated hexagons. Furthermore we characterize when this construction gives convex hexagons. If time permits we present how we use intersection theory to prove that a very general quartic is adjoint to exactly 864 heptagons.

In this work we combine and generalize earlier works of Giunti-Höfer-Velazquez (on the homogenization of the Poisson equation in random critically perforated domains) and Ansini-Braides (on a variational approach for the more general nonlinear vectorial problem in the periodic setting), each one in the direction of the other. Namely, we show that under similar general assumptions on the geometry of the random perforations as the ones posed in the work of Giunti-Höfer and Velazquez, the stochastic analogue of the result of Ansini-Braides holds true, with an average deterministic nonlinear capacitary-term appearing in the Γ-limit. This is joint, ongoing work with Caterina Zeppieri and Lucia Scardia.

We prove that every sum of squares in the rational function field in two variables K(X, Y ) over a hereditarily pythagorean field K is a sum of 8 squares. More precisely, we show that the Pythagoras number of every finite extension of K(X) is at most 5. The main ingredients of the proof are a local-global principle for quadratic forms over function fields in one variable over fields carrying a complete valuation of rank one due to Vlerë Mehmeti and a valuation theoretic characterization of hereditarily pythagorean fields due to Ludwig Bröcker. This is joint work with Nicolas Daans, David Grimm, Gonzalo Manzano Flores and Marco Zaninelli.

A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. This talk connects matroid theory with algebra via the so-called von Staudt constructions. These are combinatorial gadgets to encode polynomials in matroids. I will discuss generalized matroid representations as arrangements over division rings, subspace arrangements and as entropy functions together with their relation to group theory. As an application this yields a proof that the conditional independence implication problem from information theory is undecidable. Based on joint work with Rudi Pendavingh and Geva Yashfe.

The homogenization of a conductive medium randomly perforated with inclusions of infi- nite conductivity is a well-known problem thanks to the work of Vassili Zhikov. However, the existence of an effective model is shown under assumptions on the interparticle dis- tance, which prevents the study of clusters and dense setting. In this talk of stochastic homogenization, we will provide a relaxed criterion ensuring homogenization relying on ideas from network approximation. This is joint work with David Gérard-Varet from Université Paris-Cité

Motivated by the appearance of complex microstructures in the modelling of shape-memory alloys, we study the energy scaling behaviour of some N-well problems with surface energy given by a singular higher-order term. In the case of absence of gauge invariances (e.g. with respect to the action of SO(n) or Skew(n)), we provide an ansatz-free lower bound which relies on a bootstrap argument in Fourier space and gives evidence of the higher order of lamination involved. The upper bound is provided by iterated branching constructions. In the end, we show how a similar approach can be used in the determination of a lower bound for a more realistic model, namely the geometrically linearized cubic-to-tetragonal phase transition, in which a second order lamination is forced by the presence of affine boundary conditions. This is a joint work with Angkana Rüland.

The topic of the talk evolves around the following question: which polynomials of degree $m$ in $n$ variables are the Taylor polynomials of a rational function of degree $(d,e)$? In other words, what conditions must satisfy the coefficients of a polynomial to let it be the Taylor expansion of a rational function? For every choice of the parameters $(n,d,e,m)$ this defines a projective variety in the set of degree m polynomials, that we call the Taylor variety. In the talk I will present some results concerning the dimension theory and remarkable properties of Taylor varieties. The main motivation comes from numerical analysis, precisely, from the so-called Padé approximation problem, where the polynomial is the Taylor polynomial of a regular function around the origin, and one looks for a rational function of low degree that has the same Taylor polynomial. Joint work with A. Conca, G. Ottaviani and B. Sturmfels.

We study the behavior of harmonic functions on the infinite cluster in supercritical Bernoulli bond percolation. Most of the results which have been proved on these functions assert that they behave similarly to the harmonic functions on the lattice Zd (a typical example is the Liouville theorem on the percolation cluster). In this talk, we will be interested in the differences between harmonic functions on the lattice and on the percolation cluster through some specific properties. In this direction, we prove that there cannot exist a Lipschitz harmonic function on the percolation cluster. This property is false on the lattice as the function x -> x_1 is both 1-Lipschitz and harmonic. We will review some of the known results on harmonic functions on the percolation cluster and discuss some ideas of the proof. This is joint work with A. Bou-Rabee and W. Cooperman.

Ist q eine quadratische Form auf einem F-Vektorraum V und a ein Element des Körpers, so induziert q einen Graphen G=(V, E), wobei zwei Knoten x und y genau dann verbunden sind, wenn q(x-y)=a gilt. Graphen dieser Art sind Gegenstand zahlreicher kombinatorischer Untersuchungen. Das populärste Problem ist das Hadwiger–Nelson Problem, welches nach der Färbungszahl des Graphen der quadratischen Form <1, 1> auf den reellen Zahlen fragt. Die Bestimmung der Färbungszahl hängt zusammen mit dem Finden von Cliquen, d.h. Mengen von paarweise adjazenten Knoten des Graphen. Dieses Problem übertragen wir in die Sprache quadratischer Formen und skizzieren die Lösung im Fall von quadratischen Formen über endlicher Körpern sowie den rationalen Zahlen.

A classical result of Gage-Hamilton and Grayson asserts that any smooth, closed and simple curve in the plane evolving by mean curvature flow (MCF) shrinks to a point in finite time and becomes circular in the process. We establish a weak-strong stability result beyond the time of such circular topology change: any weak (i.e., BV) solution of planar multiphase MCF starting sufficiently close to a smooth, closed and simple curve evolving by MCF has to stay close to it for all times. Previous weak-strong stability results of this form are limited to time horizons before the first topology change of the strong solution. The reason is that the time-dependent constant in the associated Gronwall estimate is borderline non-integrable. We overcome this issue by developing a weak-strong stability theory for circular topology change up to dynamic shift, which amounts to dynamically adapting the strong solution to the weak solution to a degree which takes care of the leading-order non-integrable contributions in the Gronwall estimate. This strategy is in a sense reminiscent of the recently developed theory of L^2-stability up to shift for solutions of conservation laws close to shock solutions due to Vasseur and co-authors.

Im Zuge der fortschreitenden Digitalisierung halten Lernplattformen, die neben Aufgaben auch automatisches Feedback bereitstellen, verstärkt Einzug in den Schulalltag. Im ersten Teil des Vortrags wird eine vergleichende Analyse von Aufgaben aus Lernplattformen und Schulbüchern hinsichtlich des kognitiven Aktivierungspotenzials sowie der einbezogenen Visualisierungen vorgestellt. Im zweiten Teil des Vortrages wird ein Forschungsprojekt präsentiert, indem der Frage nachgegangen wird, inwiefern Feedback auf Lernplattformen durch eine Kombination von Selbst-Assessment und automatischem Assessment stärker lernendenzentriert gestaltet werden kann.

The general question in this talk is whether eigenvalues of operators defined on geometric spaces converge as these spaces converge to a limit space. Because the operators that we consider are often nonlinear, we need proper generalizations of eigenvalues, such as Krasnoselskii eigenvalues. I'll describe how to obtain continuity for eigenvalues of the Laplace operator defined on CD(K,N) spaces under measured Gromov-Hausdorff convergence, and I will explain how to get semicontinuity of Krasnoselskii eigenvalues under volume flat convergence. The talk contains results from joint works with Luigi Ambrosio, Shouhei Honda, Jeff Jauregui and Raquel Perales.

International wird das Beweisen als eine der zentralen Hürden im Übergang von der Schule zur Hochschule in mathematikhaltigen Studiengängen betrachtet. In dem Vortrag werden verschiedene empirische Ergebnisse zur Thematik dargelegt, um dieses Problemfeld genauer beschreiben zu können. Es stellt sich weiterführend zum einen die Frage, welche Implikationen sich hieraus für die Hochschuldidaktik der Mathematik ergeben. Zum anderen müssen Überlegungen angestellt werden, wie die Kompetenz „mathematisch Argumentieren“ in der Schulmathematik gefördert werden kann.

We give an overview on the recent results concerning the compressible Navier-Stokes system subject to stochastic forcing. The theory is based on the concept of dissipative martingale solutions. These solutions are weak in the probabilistic sense as well as in the analytical sense. Moreover, the evolution of the energy can be controlled in terms of the initial energy. We present results on the behavior of solutions in short-time (where unique smooth solutions exists) as well as in the long term (existence of stationary solutions). Finally, we investigate the asymptotics with respect to several parameters of the model based on the energy inequality.

Eye-Tracking ist eine vielversprechende Forschungsmethode in der Mathematikdidaktik, die zunehmend an Bedeutung gewinnt. Das Interesse an Eye-Tracking wird durch die technischen Entwicklungen der letzten Jahre verstärkt, durch die Eye-Tracking unter anderem für den Einsatz mit Schüler*innen leichter anwendbar und präziser geworden ist. Jedoch ist die Auswertung von Eye-Tracking Daten sehr aufwendig, vor allem bei qualitativen Analysen von Vorgehensweisen von Schüler*innen und bei Studien mit großen Teilnehmendenzahlen. Künstliche Intelligenz (KI) bietet hier Möglichkeiten der Unterstützung sowie neue Perspektiven. Im Vortrag werden verschiedene Studien und Projekte – etwa zu Vorgehensweisen bei der Anzahlerfassung und am Zahlenstrahl – vorgestellt, in denen KI im Sinne von unsupervised machine learning genutzt wurde, um Vorgehensweisen von Kindern zu identifizieren. Der Vortrag stellt darüber hinaus Möglichkeiten des supervised machine learning für die mathematikdidaktische Forschung dar.

We consider the dynamics of the repulsive Vlasov-Poisson equations near a point charge, addressing the question of its stability: For solutions which start as small, smooth and suitably localized perturbations of a point charge we capture the precise asymptotic behavior. Building on optimal decay estimates, this is revealed to be a modified scattering dynamic. Our analysis relies on the Hamiltonian/symplectic structure of the equations, and makes use of an exact integration of the linearized equation through angle-action coordinates. This is joint work with Jiaqi Yang (ICERM) and Benoit Pausader (Brown University).

A kinetic description of a plasma in external and self-consistent fields is given by the Vlasov equation for the particle distribution function coupled to Poisson’s equation for the self-consistent fields. The numerical simulation of this model is challenging due to the high-dimensionality of the phase-space. In this talk, I will discuss the challenges for a parallel implementation of such a six-dimensional problem and possible solutions like efficient data structures and how to improve data locality.

Vor dem Hintergrund einer zunehmenden qualifikationsheterogenen Lehrerschaft wird in diesem Vortrag ein am Deutschen Zentrum für Lehrebildung Mathematik (DZLM) entwickeltes Konzept einer Stochastik-Fortbildung für Lehrpersonen, die Mathematik in der Grundschule unterrichten vorgestellt. In fünf Kursdurchführungen wurde dieses Konzept realisiert und untersucht. Im Vortrag werden Ergebnissen von Lernprozessen bezüglich eines fachlichen und fachdidaktischen Wissenszuwachses interpretiert. Zudem wird betrachtet, in wieweit eine qualifikationsheterogene Zusammensetzung der Kursgruppen die Durchführung und das Lernen der Lehrpersonen beeinflusste. Es werden Einblicke in Entwicklungsprozesse unterschiedlich qualifizierter Lehrpersonen in dieser fachinhaltlich orientierten Fortbildung gewährt. Abschließend werden mögliche Impulse für Unterrichtsentwicklungsprozesse durch eine Einbindung des PLG-Ansatzes in das Kurskonzept und Erprobungen in den Praxisphasen diskutiert.

We present a machine learning approach for developing stochastic parametrizations of subgrid fluxes in finite-volume discretizations of PDEs expressed in flux form. In particular, we utilize Generative Adversarial Networks (GANs) to parametrize subgrid fluxes in equations for coarse variables defined via spatial filtering. We discuss how to construct and train the network and demonstrate that our approach reproduces statistical properties of coarse variables in fully resolved simulations. Moreover, we also demonstrate that GANs are robust and can be utilized without re-training in new regimes (e.g. changes in forcing). We use the Burgers equation to illustrate our approach.

Der von Markus Rost und (unabhängig) von Alexander Vishik bewiesene Satz, daß Rost Nilpotenz für projektive Quadriken gilt, spielt eine wichtige Rolle in der sogenannten geometrischen Theorie der quadratischen Formen. Es wird vermutete, daß Rost Nilpotenz für alle glatten projektiven Varietäten gilt. In dem Vortrag soll zuerst Rost Nilpotenz erklärt werden, und dann über einige dazu bekannte Resultate berichtet werden. Im Anschluss wird eine Verallgemeinerung, die Rost Nilpotenz impliziert, diskutiert. (Letzteres ist ein gemeinsames Projekt mit Alexander Vishik.)

Im allgemeinen ist über die kohomologischen Invarianten von endlichen oder algebraischen Gruppe nur wenig bekannt. Eine Ausnahme bilden die Invarianten vom Grad 2, also die Invarianten mit Werten in der Brauergruppe, diese wurden in den letzten Jahren berechnet. Im Vortrag wird zuerst diese Berechnung angegeben, und dann ein elementarer Beweis im Falle der endlichen Gruppen skizziert.

The theory of optimal transport provides a rich mathematical background with connection to various fields in applied mathematics, among them convex analysis, duality theory, and partial differential equations. Moreover, the algorithmic treatment of optimal transport problems has made significant progress in the recent past. This is of major importance, since optimal transport problems arise in various applications such as image processing and urban planning. The summer school aims to give a broad introduction into the field of optimal transport, from theory over numerics to applications. It is primarily intended for the doctoral and postdoctoral students of the DFG SPP1962, however other interested parties may also apply to register.

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.

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.

In this talk we will consider random walks on supercritical Galton-Watson trees with random conductances. That is, given a Galton-Watson tree, we assign to each edge a positive random weight (conductance) and the random walk traverses an edge with a probability proportional to its conductance. On these trees, the random walk is transient and the distance of the walker to the root satisfies a law of large numbers with limit the speed of the walk. We show that the distance of the walker to the root satisfies a functional central limit theorem under the annealed law. In particular, we are interested how the variance changes when the conductances on a positive fraction of edges tend to zero.

Kontextuelle Graphen stellen einen funktionalen Zusammenhang dar, der bezogen ist auf eine realweltliche Situation, aus der die Daten stammen. Sie sind oft unregelmäßig und nicht durch einen Term beschreibbar. Ihre Interpretation ist herausfordernd. Um zu verstehen, wie Schüler*innen Graphen interpretieren, benötigt man Einblicke in ihre kognitiven Prozesse während der Interpretation. Eine aussichtsreiche Methode, um Erkenntnisse über kognitive Prozesse und individuelle Vorgehensweisen zu erlangen, ist Eye-Tracking. Die Interpretation von Blickbewegungen ist jedoch nicht trivial. Im Vortrag werden methodologische Überlegungen angestellt zur Untersuchung von Blickbewegungen bei der Interpretation kontextueller Graphen. Anschließend werden diese Erkenntnisse genutzt, um anhand von Blickmustern zu untersuchen, wie Neuntklässler*innen die Änderung kontextueller Graphen erfassen. Diese Vorgehensweisen werden in Hinblick auf mögliche Einflussfaktoren der genutzten Vorgehensweisen diskutiert.

We consider the sharp interface limit of a Navier-Stokes/Allen-Cahn system, when a parameter $\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. In dependence on the mobility coefficient in the Allen-Cahn equation in dependence on $\varepsilon$ different limit systems or non-convergence can occur. In the case that the mobility is proportional to the square root of $\varepsilon$ solutions convergence to a solutions of a classical sharp interface model on short times for well-prepared and sufficiently smooth initial data. To this end we construct a suitable approximation of the solution of the Navier-Stokes/Allen-Cahn system using an expansion with half-integer powers of $\varepsilon$ and suitable solutions of a linearized limit system. Then the difference of approximate and exact solution is estimated with the aid of a suitable spectral estimate of the linearized Allen-Cahn operator. This is a joint work with Mingwen Fei (Anhui Normal University, China), and Maximilian Moser (ISTA Klosterneuburg, Austria).

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)

Die Implementation früher mathematischer Bildung in Kindertagesstätten kann durch Fortbildungen für frühpädagogische Fachkräfte sowie Lehrkräfte an Fachschulen unterstützt werden. Werden Multiplikator*innen qualifiziert, um wiederrum Fortbildungen für frühpädagogische Fachkräfte bzw. Lehrkräfte zu gestalten, können mit diesen Fortbildungen eine Vielzahl von Fach- und Lehrkräften erreicht werden. Fraglich ist jedoch, wie erfolgreich dieser Scaling Up-Ansatz mit Blick auf die inhaltiche Tiefe und Breite ist. Im Vortrag werden ausgewählte Erkenntnisse aus den Projekten Projekten EmMaM und EmMaM-FS dargestellt. Dabei wird sowohl die Umsetzung der Fortbildungen durch die Multiplikator*innen als auch die Wirksamkeit dieser Fortbildungen betrachtet. Aus den Erkenntnissen werden erste Implikationen für Folgeprojekte abgeleitet und zur Diskussion gestellt.

Minimal discrete energy problems arise in a variety of scientific contexts – such as crystallography, nanotechnology, information theory, and viral morphology, to name but a few. Our goal is to analyze the structure of configurations generated by optimal (and near optimal)-point configurations that minimize the Riesz s-energy over a sphere in Euclidean space R^d and, more generally, over a bounded manifold. The Riesz s-energy potential, which is a generalization of the Coulomb potential, is simply given by 1/r^s, where r denotes the distance between pairs of points. We show how such potentials for s>d and their minimizing point configurations are ideal for use in sampling surfaces.
Connections to the breakthrough results by Field's medalist M. Viazovska and her collaborators on best-packing and universal optimality in 8 and 24 dimensions will be discussed. Finally we analyze the minimization of a 'k-nearest neighbor' truncated version of Riesz energy that reduces the order N^2 computation for energy minimization to order N log N, while preserving global and local properties.

We consider a weighted discrete graph over , i.e. is a countable discrete set, is a function on which induces a measure and 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 and a subset of , the norm of the solution of the heat equation at time can be bounded by the portions of the solution on up to time 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).

The talk will present advances on efficient numerical methods for solving fluid flows with two focus areas, the solution of incompressible turbulent flow and high-Mach number flows with shocks. For turbulent incompressible flows, high-order discontinuous Galerkin methods are used. Based on mathematical ingredients that ensure robustness for marginally resolved flow structures, the talk will present techniques to enforce the incompressibility constraint as well as inter-element continuity of the velocity field, using techniques of H(div)-conforming Raviart-Thomas finite elements. For the efficient implementation, matrix-free operator evaluation with fast evaluation of the finite-element integrals is developed, using the concept of sum factorization. This method utilizes the properties of modern multi-core processors and GPUs in an optimal way by replacing memory-intensive sparse matrices of traditional solvers by redundant computations of integrals with a higher proportion of arithmetic work and less memory transfer. In the second part of my talk, I will discuss the efficient implementation of robust methods for the compressible Euler and Navier-Stokes equations. These methods preserve the invariant domain of the Euler equation and are therefore provably stable also near shocks, at the cost of expensive operations to evaluate limiters and shock indicators along stencils. For the viscous effects, matrix-free multigrid solvers are used.

Die Lehrkräftekognitionsforschung hat in den letzten 15 Jahren entlang der Konstruktkette professionelles Wissen – Teacher Noticing – professionelle Handlungskompetenz eine solide Basis empirischer Befunde hervorgebracht. Diese beziehen sich vor allem auf die Natur, die Struktur, teilweise auch die Entwicklung der zugehörigen Bereiche. Der Vortrag gibt einen Überblick zu theoretischen Grundlagen und ausgewählte Ergebnisse aus verschiedenen empirischen Studien (u.a. TaiGer Noticing, ELMaWi) in Hinblick auf Ableitungen für die Praxis der Lehrkräfteaus- und -fortbildung.

Large-scale oceanic flows can often be described by quasi-geostrophic models. These are applicable if the Rossby number is small, as is the case in the earth’s atmosphere and oceans. Geostrophy means that the Coriolis force is balanced by the horizontal pressure gradient, thus allowing for additional simplifications compared to the (hydrostatic) Navier–Stokes equations. In this talk, I will illustrate the derivation of such systems, discuss the imposition of boundary conditions and show preliminary numerical results based on global spectral methods. The models and solvers we develop are interesting for a number of applications, in which the effects of bottom topography have been neglected thus far. We hope to fill these gaps in the course of this project. First, we plan to study how topography affects the spin-up of a stratified ocean at rest, for which there is a theoretical foundation in the case of flat bottoms.